Determinants are very important in almost every part of mathematics that I know. I wish I understood them better. But working directly with matrix determinants is very cumbersome, computation-intensive, and often unenlightening. So indeed they are not taught as much as they used to be. For pure mathematicians they have been replaced by exterior algebra, which is a much more elegant setting. And, as Spiro mentions, geometers work with signed volume forms, which turn into determinants when written with respect to a basis.
Even in computational mathematics, there is something called geometric algebra, which uses exterior, quaternionic and Clifford algebras to encode all the algorithms in an elegant and yet computationally efficient manner. I'm under the impression that video game programmers find this very useful.
