Let
$V$
be a vector space over a field
$F$
, let
$r$
be a positive integer.and let
$L$
be a multilinear function (linear in each argument)
$L:V^r \rightarrow F$
.
$L$
is an alternating form if
$L=0$
whenever two arguments are the same, and
$L$
changes sign if conservative arguments are interchanged.
Let
$L$
be an r - linear form on
$V$
and let
$\sigma$
be a permutation of
$1, \; 2,..., \; r$
- there are
$n!$
such permutations). If
$L$
is alternating the sign of
$L$
is defined by the number of transpositions of
$1, \; 2,..., \; r$
needed to return the set to the natural order. Suppose that numbers
$1, \; 2,..., \; r$
are rearranged as
$a_1, \; a_2,..., \; a_r$
then
$L(\alpha_1 , \; \alpha_2 , \;..., \; \alpha_r)=(-1)^{e_{a_1a_2...a_r}}L(\alpha_{a1} , \; \alpha_{a_2} , \;..., \; \alpha_{a_r})$

where
$e_{a_1a_2...a_r}$
is the number of transitions needed to return the set
$a_1, \; a_2,..., \; a_r$
to the natural order.
Write
$(-1)^{e_{a_1a_2...a_r}}=sgn( \sigma )$
then
$L(\alpha_1 , \; \alpha_2 , \;..., \; \alpha_r)=sgn( \sigma )L(\alpha_{a1} , \; \alpha_{a_2} , \;..., \; \alpha_{a_r})$

The set of all multilinear functions forms a subspace
$M^r(V)$
. For each
$L \in M^r$
define
$\prod_{\sigma } L \in M^r(V)$
as
$\prod_{\sigma } L= \sum_{\sigma} sgn(\sigma ) L$
.
$\prod_{\sigma } L$
is a linear transformation from
$M^r(V)$
onto
$A^r(V)$
- the set of alternating r forms on
$V$
.