I'm going to make a couple assumptions that you need an ATX motherboard, as well as live in the US and are using this PC for gaming. Here is a recommendation based on those assumptions.
Bf4 takes advantage of many cores so fx 8xxx series will perform better than i3 where as fx 6300 will also perform better than i3 , but for skyrim situation is different, not to mention you can overclock amd processors , but i3 can not be overclocked
In BF4 fx 8350 is on par with i5 or is a little better and in multithreaded optimized games and apps its on par with i7, but your right currently for gaming i5 is the best to go as very few games actually take advantage if more than 4 cores ,
This is pushing the budget but you get an extremely fast i5 for under $200 and a mobo that supports USB 3.0 (which is good if your case has usb3.0 in front of case).