Let \(F\) be a field and let \(V_{1},\ldots ,V_{n}\) and \(W\) be vector spaces over \(F\). A multilinear map. \[\Phi \colon V^{n}=\underbrace {V\times \cdots \times V} _{n-{\text{times}}}\longrightarrow W\,\]
is said to be alternating, if \(\Phi (v)=0,\) whenever in the tuple \(v=(v_{1},\ldots ,v_{n})\) to different entries are equal, i.e. \(v_{i}=v_{j}\) for a pair \(i\neq j\).