secondly I would get a better videocard and recommend Nvidia. Remember clockspeed on the processor is more important than VRAM.
This build could run Skyrim, but if you are hoping for it to run it on high settings, no it would not. I would guess probably medium settings.
I guess my main point is if you want a gaming PC you may need to spend more.
I would have to disagree on some points of this message. An i5 is no doubt awesome, but $100 more is quite a bit of cash for some people.
AMD has good cards as well, and the only reason I recommend them more often than not is because of their much more competitive pricing. The only thing that is better than the 7770 Nvidia-wise around that price is the 650Ti, which does not have enough of an advantage over the 7770 to justify its cost.
Skyrim would probably run medium-high ish though, you are right.