Yup, that's how it goes. You don't even need the full 10 if places remain untaken by the end of the round.
Important to note that only the top unclaimed place can be scored at the end of the round. If first has already been taken and discarded, and at the end plyr A has 5 rolls and plyr B has 3 rolls, only plyr A scores 5 for 2nd. Plyr B doesn't also score 2 for 3rd. Likewise, if plyr A put down a 5 and 3 (8 total), and there is only one other uramaki, he should try not to take it because he still scores 8 at the end, and plyr B scores zero. This is a strong blocking technique in 2p.
(in fact, I think a little too strong in 2p only. So, for 2p only, we play a race to 8, and score 7, 4, 2 for 1st, 2nd, 3rd, respectively)