I use the simple Pace skull caps year 'round to 1) keep the inside of my helmet clean, 2) keep sweat out of my eyes I still have hair but I'm sold on those caps and I don't think losing it will make much difference -helmet burn-wise.
(A trick to keep sunscreen out of your eyes - Aloe Gator 40SPD Gel. That stuff does not run and if it does get in your eyes, burns very little. I apply it over my eyelids. I don't sweat it getting on my eyelashes and rarely have any issues. I only use it around and above my eyes as doesn't wash off easily and every bug that lands in it stays.