Multilinear mapping/Alternating/Linear subspace/Exercise

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be vector spaces over ${\displaystyle {}K}$, and ${\displaystyle {}n\in \mathbb {N} }$. Show that the set of all alternating mappings (denoted by ${\displaystyle {}\operatorname {Alt} _{K}^{n}{\left(V,W\right)}}$) is a linear subspace of ${\displaystyle {}\operatorname {Mult} _{K}{\left(V,\ldots ,V;W\right)}}$ (where the vector space ${\displaystyle {}V}$ appears ${\displaystyle {}n}$-fold).