Tensor product/Wedge product/Fact

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

Then there exists a canonical

V\otimes _{}\cdots \otimes _{}V\longrightarrow V\wedge \ldots \wedge V,v_{1}\otimes _{}\cdots \otimes _{}v_{k}\longmapsto v_{1}\wedge \ldots \wedge v_{k}.