When skills reach Level 20

When the skills for staff reach level 20, that is the maximum they get to, and you get no other bonuses or skill improvement?

Are you talking about staff or drivers? If you're talking about drivers then yes the maximum is 20 and no, there are no bonuses. Atrophy is always there though, so the stats don't stay at 20. You eventually have to retrain everything.

The maximum level for staff is also 20.

I have noticed that if you click on a skill that is already at 20, then it tells you as such but does not allow you to train one of the other skills, effectively losing a day of skill trianing.

Oh you are talking about the team staff.
You get no training after 20 but the bonus (commercial: +10% money next race from sponsors, engineering: -10% cost for changing parts, design: +1% on next seasons design, doctor: driver health +1 and morale +1)