My Xbox 360 wired controller is going bad.

Generic PC gamepads that can't be played on any consoles are much cheaper than X360 wired controllers.

If I plugged one of the in, would Skyrim recognize it the same as it would an X360 controller?

Aware from very cheap ones. The may work or even not. !!!

If you want a good branded product and way cheaper than the XBOX 360 wired controller the go with this one. (it's almost half the price. As i see the price has dropped recently...)

http://www.amazon.com/Logitech-940-000110-Gamepad-F310/dp/B003VAHYQY/ref=sr_1_1?ie=UTF8&qid=1446317815&sr=8-1&keywords=logitech+f310

....and works out of the box with W7, W8 (8.1) and W10.

I own the F310, and i can recommend this one. !!! (So just go and grab one)

If it supports Xinput, then it should work using the same system as the XBOX 360 controller, otherwise you will have to manually rebind input to keyboard inputs using third party applications, for example 'joytokey' or 'xpadder'. Most controllers should support Xinput.

Isn't Xpadder unpreferred?

I tried using Xpadder to play Oblivion on a gameplad, and that didn't go well. It just didn't simulate the X360 controls.

Case in point: When I try to navigate the menu, I had to use the right thumbstick to mvoe my cursor over the icon I wanted to click, and then use the right trigger to click on it, whereas on the X360 itself, I could just use the RT and LT buttons to shift between sub-menues, the left thumbstick to highlight items, and the A button to select, which was much more intuitive, but I couldn't get that on the PC.

With Skyrim, if I plug in the X360 controller, then it works perfectly; it's a seemless adoption of the X360 control style.

Obviously you want native Xinput support, and not have to rebind inputs.