"It's primarily an open-world fantasy action-adventure oriented first-person stat based role playing game (like pretty much the only series in this category)".
By that definition, it's pretty much the Elder Scrolls, since Fallout 3 and New Vegas don't count because they're not fantasy, and it rules out most cometitors because there aren't really all that many first person RPGs these days, and even fewer if it needs to be open. Other games just don't qualify, and one series isn't really enough to warrant a genre. Though if I wanted to describe specifically what the Elder Scrolls is, I'd call it a (primarily) first person fantasy open world RPG, I'd go into more detail if necessary, of course, but that seems like a good description of the series as a whole.
As to the RPG as a whole, it's pretty hard for me to give a concrete definition of the genre, perhaps because there isn't really one formula all RPGs follow. But they do tend to have some common themes that most RPGs tend to follow. Generally speaking, an RPG must at the very least allow you to develop your character through various skills, stats or abilities that help to define your character, typically, these abilities improve as you gain experience, with your character starting out quite unexperienced but become better as the game progresses, learning new abilities and being able to overcome greater challenges as a result. The exact system for this progression isn't the same for every game, of course, but some means of character progression should be there, and generally, the player has some control over how the character's abilities develop, allowing the player to choose what sort of character to play, within the limit of the game. This does not make every game that has skills automatically an RPG, as there have been some games in other genres which have included light RPG elements, but don't put enough emphasis on them to really be called RPGs. The gameplay of RPGs can vary considerably but they usually have some common aspects, RPGs will typically involve a fair amount of questing and exploration, with players needing to complete a variety of quests to complete the game, often, they will also have side quests which aren't necessary to complete the main story but may give players useful rewards if they are completed. RPGs also typically have a fairly large amount of characters the player can interact with, some of these characters may be enemies, some may be allies, or some may be neutral to the player, sometimes, their relationship with the player will be defined by the player's actions. Some of these NPCs may offer the player quests, or advance quests, whereas some may provide services or just provide dialog that isn't necessary for quests, but may or may not still be interesting to listen to. Also, one thing I expect to see in an RPG is an inventory system, this is important, as aside from skills, the equipment you choose can also define your character. Warriors for example often favor heavy armor and might use swords, axes, maces, or other such weapons, mages traditionally wear robes and carry staves. Of course, depending on the mechanics of the game, the player may have more choice in terms of what kind of equipment a character uses than others. The Elder Scrolls for example generally imposes no limitations on what certain character types can use in terms of equipment, you can wear any armor and carry any weapon you want, you'll just be less effective with it if your not skilled in it. RPGs are also likely to feature elements of choice, how many choices the player gets and how strong the concequences they bring with them are varies between games, as well as the sort of choices the player gets. Often, these are simple choices between good and evil, but some games might take a more complex approach to this and face the player with difficult choices where all options may have their merits, or the best outcome the player can get is merely the lesser evil, it depends a lot on the nature of the game, and also the general feel that the designers are going for. RPGs don't necessarily have to take place in a fantasy setting, though a lot of them are. Some are instead set in a science fiction setting, in which case, the gameplay mechanics may be adjusted to fit the setting, for example, whereas fantasy RPGs typically allow players to use magic of some form, science-fiction RPGs might either replace it with either something similar but derived from pseudo-science rather than supernatural forces, or may just do away with similar mechanics entirely, like Fallout, for example. RPGs also often allow players to customize their character at the start of the game, though some games avoid this, instead giving players a premade character but allowing players to choose how to play said character. While I generally prefer being able to create my own character, but I've also encountered some RPGs that do not have such a feature which I've also enjoyed, so really, it depends on the game.
Of course, like I said, those are just common aspects that tend to be present in RPGs, that is not a strict definition of an RPG, like I said, it would not be easy for me to do so perhaps because the RPG genre is a pretty varied genre of games that are all connected by some common elements, not every genre has that kind of variety. Take first person shooters, for example, all of them, as their name would imply, follow a certain framework, which is that they are played from a first person perspective, and usually, you have to shoot enemies. The settings may differ, though often, they take place in modern times, the reletively recent past, or some point in the future, considering that technically a game where melee combat is the primary focus shouldn't be called a shooter, so games set in the Middle Ages or your typical psuedo-Medieval fantasy setting might not work so well. There may be some differences in terms of gameplay mechanics, the openness of maps (Some games, like Call of Duty, have levels that are for the most part pretty linear, whereas some games have somewhat more open level design.) some may also blend in elements of other genres, such as games that include RPG elements as mentioned above. Such as Biosock (Includes an element of character progression in the form of Plasmids.) and S.T.A.L.K.E.R. (It has an inventory system, quests, and while the world isn't completely linear, it's more open than most first person shooters.) But even so, they don't stray from the basic framework of having a first person game focusing on action, and with said action usually be resolved by shooting targets, but the RPG genre is not quite so easy to define. I think one of the problems is that unlike some genres, the name of the genre does not really accurately describe what it's about. Aside from the above mentioned first person shooters, in other genres, the name gives you a pretty good idea of what the game is about. But if you look at the name "role-playing game" literally, it could technically mean any game in which you play as an actual character, as you have to play a role in such games. In Half-Life, you play the role of Gordon Freeman, in a real time strategy game, you play the role of a military commander, usually, in each GTA game, you play the role of whoever the main character is. But it would be foolish to call any game that the label could technically describe an RPG, as then there would be no meaning to using it as the name of a genre. A genre serves as a label that gives us a basic idea of what you can expect from a particular work, be it a game, a movie, a song, a book, or whatever. In video games, genres are usually based on the overall gameplay experience, rather than the actual content of the work like in most forms of non-interactive fiction. If a game is defined as a certain genre, we expect certain common aspects that define said genre from it, it also helps us find games which may offer the sort of experience we're looking for. When I first got Morrowind, I chose it because at the time, I wanted to get a new RPG, and thus, someone was able to recommend an RPG to me. Obviously, genre alone is not enough to judge a game, because just because I like a certain genre as a whole doesn't mean I will like everything in it, but it's something to start with, and the genre loses its meaning when there aren't certain common elements we can expect from a work in said genre. And for those, when it comes to RPGs, the elements I had mentioned before are some of the ones I expect to see.
So to conclude a wall of text post, an RPG is not something I can easily define, but I have some idea of what I aught to expect when a game is called an RPG, and if I play a game, I can probably tell you if I'd call it an RPG or not even if I went into it without knowing what genre it was. Also, there aren't really any particular elements I'd say an RPG shouldn't have so long as they don't detract from what makes it what it is, and really, as with any genre, what a game should have should depend on what the developers are going for with a game, a genre doesn't need to impose strict limitations on what a game can be.