Why is this a good definition of cross product

Question

In do Carmo's Differential Geometry of Curves and Surfaces he defines the cross product of two vectors $u,v \in \mathbb{R}^3$ as the unique vector such that for every $w \in \mathbb{R}^3$, $(u \wedge v)\cdot w=det(u|v|w)$ where the notation indicates that the determinant is of the matrix having as columns the components of each vector. The author then goes on to prove the useful properties of the cross product.
My question is, why is this a good definition? Because, when I was first introducted to the cross product, I was only told that it was a vector perpendicular to both the members of the product and of which length obeyed a particular restriction. How does this "determinant" definition capture this other geometric definition. Is it merely that we found that it was "computationally" the same? That is, someone eventually computed that determinant and saw that it gave the same value as the dot product of the cross product with a general vector? Or is there more to it? Thanks!