Wedge product/Universal property/Alternating forms and linear forms/Fact

Let ${}K$ be a field, let ${}V$ be a ${}K$-vector space, and ${}n\in \mathbb {N}$.

Then there exists a

natural isomorphism

${\left(\bigwedge ^{n}V\right)}^{*}\longrightarrow \operatorname {Alt} ^{n}(V,K),\psi \longmapsto ((v_{1},\ldots ,v_{n})\mapsto \psi (v_{1}\wedge \ldots \wedge v_{n})).$

Proof 1, 2, Write another proof