I use shampoo on my face / beard and, seeing as it was formulated for your head (and scalp), I guess shampoo ought to be the ideal thing (even if we don't have hair)?!
I think what's more important is moisturising and protecting one's scalp - it is so easy to get burned up there and the skin is so delicate. I pretty much use sunscreen from April onwards even here in the UK. It only takes 20 minutes and I can be sore, 30 and I'm burnt.
I get asked quite a lot if I still have to use shampoo, mostly by the kids I teach. I always tell the truth and say "Yes... in order to be clean!" What I don't admit to is how long a bottle of shampoo actually lasts, which is measured in many months - I use so little at a time that I'm almost embarrassed by how old the bottle looks when I finally run out.