Couldn't put together a decent setup at $300, but managed to squeeze just enough at $354
http://pcpartpicker.com/p/3giJG
That's the bear minimum if you want to play Skyrim, you can see from the following review that the GTX 650 I picked for you can even handle it at 1920x1080 at high details (albeit barely), so it should be fine at 1360 x 768.
http://www.bit-tech.net/hardware/graphics/2012/10/04/evga-geforce-gtx-650-1gb-review/5
The CPU is also the bear minimum for gaming. Obviously you could save a few dollars on the cpu / gpu and still manage to play, but keep in mind that every dollar you save on any of these components will dramatically reduce the performance because the cheaper parts are very low-end.
Keep in mind that the motherboard does not support USB 3.0, if you want that you'll have to add around $13 for the Gigabyte GA-F2A75M-HD2.
I feel that the only components you'll be able to save money on without losing tons of performance are the case, which is also the bear minimum for me because that's the cheapest one I can personally recommend, but maybe if you research a bit you'll be able to find a better deal.
You'll also be able to save $15 if you don't get the dvd writer, but I added it because I thought that you might want to watch some blockbusters on the TV using the computer.
Hope that helped, good luck.