A skew-symmetric (or antisymmetric) bilinear form is a special case of a bilinear form , namely one which is skew-symmetric in the two coordinates; that is, for all vectors and . Note that this definition only makes sense if is defined over two identical vector spaces, so we must require this in the formal definition:

a bilinear form ( a vector space over a field ) is called skew-symmetric iff

for all vectors .

Suppose that the characteristic of is not . Set in the above equation. Then for all vectors , which means that , or . Therefore, is an alternating form.

If, however, , then ; is a symmetric bilinear form.

If is finite-dimensional, then every bilinear form on can be represented by a matrix. In this case the following theorem applies:

A bilinear form is skew-symmetric iff its representing matrix is skew-symmetric. (The fact that the representing matrix is skew-symmetric is independent of the choice of representing matrix).