Alternating mapping/Swap property/Fact

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

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

is an alternating mapping.

Then

$\Phi (v_{1},\ldots ,v_{r-1},v_{r},v_{r+1},\ldots ,v_{s-1},v_{s},v_{s+1},\ldots ,v_{n})=-\Phi (v_{1},\ldots ,v_{r-1},v_{s},v_{r+1},\ldots ,v_{s-1},v_{r},v_{s+1},\ldots ,v_{n})\,.$

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

Proof, Write another proof