The geometric picture underneath is one of the things that keeps me in awe of the subject despite its seeming simplicity, and I keep getting something out of it every time I come back to it. It's a bummer since finite-dimensional linear algebra is one of a handful of mathematics topics where one can answer all the questions posed at the beginning of a course in it by the end of a course in it, so it is a pretty self-contained topic.
After learning e.g. exterior algebra (differential forms), Clifford algebra (geometric algebra in these parts), and so on, the geometric picture of the determinant as the size of an oriented volume makes deriving the algebraic formula super duper slick. Like in Clifford algebra, the formula can be proven in two or three lines. It's unfortunate that it seems like e.g. exterior algebra never get introduced sooner in the pedagogy of linear algebra or multivariable calculus because when used right they make the underlying ideas shine through beautifully. It's a bummer since exterior algebra is much simpler than it looks, though like many things in mathematics, it's takes a lot of work to make that simple idea rigorous. But unfortunately algebra in general given it's abstract nature can absolutely lobotomize the real deal geometric ideas underneath a lot of this stuff when used poorly.