An alternating multilinear form on a real vector space 
is a multilinear form
 |
(1)
|
such that 
whenever two arguments are equal.
Over 
,
this is equivalent to
 |
(2)
|
for any index 
. For example,
 |
(3)
|
is an alternating form on 
.
An alternating multilinear form on a module over a commutative ring is likewise required to vanish whenever two arguments are equal. The sign-change identity alone is insufficient in ring characteristic 2.
