Wedge product/Canonical mapping for linear mapping/Existence/Fact

Let $K$ be a field, let $V$ and $W$ be $K$-vector spaces, and let

$\phi :V⟶W$

denote a $K$-linear mapping.

Then there exists, for any

$n\in \mathbb{N}$, a $K$-linear mapping

$\stackrel{n}{\bigwedge }\phi :\stackrel{n}{\bigwedge }V⟶\stackrel{n}{\bigwedge }W,$

with $v_1\wedge \dots \wedge v_n\mapsto \phi (v_1)\wedge \dots \wedge \phi (v_n)$.

Proof, Write another proof