Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (e.g., a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module over a commutative ring.
The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
Definition.
A multilinear map of the form formula_1 is said to be alternating if it satisfies any of the following equivalent conditions:
Alternatization.
Given a multilinear map of the form formula_1, the alternating multilinear map formula_11 defined by formula_12 is said to be the alternatization of formula_13.