How can you forget you're playing a single-player game?
Skyrim is a very relaxing and soothing game. No idiots swearing and typing constantly in chat, everyone in-game is actually 'role playing', you don't see people jumping around like bunnies, shooting/casting needlessly, etc.
I like multi-player games, but I don't see how anything in Skyrim reminds me of a multi-player game so-much that I might forget I'm playing a single-player game.
I'd also recommend PC, as it runs quite well on older hardware, and can scale up the graphics if you have good hardware. Also go to Skyrim Nexus and look at some mods (more will come soon when editing tools are released). Console is a waste of money (if you have a PC as you get more value for money with this game on PC), and support the PC market by buying this on PC only.