Multilinear mapping/Alternating/Definition

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

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

is called alternating if the following holds: Whenever in ${\displaystyle {}v=(v_{1},\ldots ,v_{n})}$, two entries are identical, that is ${\displaystyle {}v_{i}=v_{j}}$ for a pair ${\displaystyle {}i\neq j}$, then

${\displaystyle {}\Phi (v)=0\,.}$