volume of parallelotopes

Question

I know that determinant indicates the volume of a parallelotopes spanned by the n vectors.

I absolutely understand that the properties of a determinant: any function $f:\mathbb{R}^{n\times n}\to\mathbb{R}$ satisfying

1.multi-linear

2.anti-symmetric

3.$f(e_1,e_2,...,e_n)=1$

than $f$ is a determinant.

However, I can't see that if a function satisfying the above constraints must be a function of volume of the parallelotopes. So how to prove volume of parallelotopes is equal to determinant of the vectors?

Answer 1

You can start with a set of arbitrary vectors and use Gram-Schmidt to orthogonalize it (that is, reduce to $e_1$,...,$e_n$).

EDIT: This way you can show that the function satisfying 1.-3. indeed gives you the volume.