I also don't have hair (by choice) but cannot stand hats. Especially billed hats. The whole reason I shave my head is I like the feel of fresh air on my head, and hats ruin that.
A couple years back, I picked up an antimicrobial "athletic bandana" (really just a breathable do rag) to wear under my bicycle and motorcycle helmets so they wouldn't get all stanky when I ride. It's EXTREMELY light, with a slightly-stretchy thin fabric on top and a thicker, more elastic fabric at the band. It even has a textured rubber strip along the front of the band to keep sweat from running down into my eyes--I thought this would be obnoxious and sticky-feeling, but I don't notice it at all.
Anyway, I like the thing so much I found myself wearing it even when I'm not riding. I'm sure it looks less than cool, but I wouldn't be walking around with a bic'd head if I was worried about that. It's comfortable, keeps the sun off, wicks sweat away, and keeps me cool--plus the antimicrobial fabric means it doesn't smell like sweat even after many uses (though I still wash it periodically).
This is the one I've got:
http://store.haloheadband.com/athletic-bandana-p/pb.htm
Edit:
Looks like they make billed caps from the same stuff, but they have adjustable closures.