Yes you can, you need a seperate key though (1 for US and 1 for EU.)
Sc2 cache etc etc gets installed in some folder, and it tells which region you're playing from, so you can't just download the EU files and play on EU server, it comes up with an error about language pack.
What you need to do is download and run this to change between US and EU language pack/servers.
http://www.teamliquid.net/forum/view...opic_id=139082
---------- Post added 2012-01-24 at 02:42 PM ----------
Which country are you in?