Determinant function/Uniquely determined/Fact

The universal property of the determinant

Let $K$ be a field and $n\in \mathbb{N}$.

Then there exists exactly one

determinant function

$\mathrm{△}:{\mathrm{Mat}}_n(K)=(K^n{)}^n⟶K$

fulfilling

$\mathrm{△}(e_1,e_2,\dots ,e_n)=1,$

where $e_i$ denote the standard vectors, namely the

determinant.

Proof, Write another proof