I actually like the idea, but it can't work this way even under the assumption that cheating is impossible.

Why? Massive deflation.

Imagine one million clients and everyone finds 5 glass claymores or similar... that's 5 million glass claymores in the auction house!
Now we have in the best case scenario one million new characters at the same time going to the auction house all wanting to buy a better sword than they can find by playing the game normally.
That's an "over production" of 4 million claymores accumulating with an other another 5 million glass claymores entering the auctionhouse...
Everybody can and will simply dump their excess items on there and you'll be able to get any equipment for extremely low prices, that would totally break the game.

What could make an auction-house work and exciting is if the game includes advanced crafting minigames where you can design clothes, weapons, furniture etc... and you can only sell those items at the AC.
Because then, players pay for the amount of work and skill that went into creating the items, which keeps the price up, keeps the market more balanced and it will make your character want to go there every now and then to see if there's something that he likes. It's not something I see happening with Skyrim though.

See this is how I thought it was a good idea. Say your character is a Mage. You don't want or use a sword you find. You can then sell that sword on the auction house for a decent amount of gold and turn around and buy something off the auction house with said gold that better suits your Mage character, like say a staff or some potions maybe. A special robe perhaps.

I don't think it would actually because unlike WoW you are not paying any money to play it so why would anyone care about in-game gold for a singleplayer only game?

Well see, that's not an issue on PC for sure: "player.additem f 10000" and you're 10,000 gold richer. But on consoles, where you aimed this idea by your own admission, you don't have a console to use "gimme gimme" commands with, and...yeah, this would indeed open the door for gold and item sellers. "TESItemzzz.com, best prices for Daedric artifacts! Xbox and PS3 services available!"