Alternating mapping/Swap property/Fact

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ denote ${\displaystyle {}K}$-vector spaces, and let ${\displaystyle {}n\in \mathbb {N} }$. Suppose that

${\displaystyle \Phi \colon V^{n}=\underbrace {V\times \cdots \times V} _{n{\text{-mal}}}\longrightarrow W}$

is an alternating mapping.

This means that if we swap two vectors, them the sign is changing.