Let $k$ be a field where $2\neq0$, and $V$ an $n$-dimensional $k$-vector space. Then there exists a unique function $\det:V^n\to k$ that has the following properties:

- $\det$ is multilinear (linear in each variable)
- $\det$ is alternating (permuting two arguments changes the sign)
- $\det(B)=1$ for some basis $B$

Without providing an explicit formula (à la Leibniz), how can existence be proved?