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

a bilinear form $B: V \times V \rightarrow K$ ($V$ a vector space over a field $K$ ) is called skew-symmetric iff

$B(x,y) = -B(y,x)$ for all vectors $x, y \in V$ .

Suppose that the characteristic of $K$ is not $2$ . Set $x=y$ in the above equation. Then $B(x,x)=-B(x,x)$ for all vectors $x \in V$ , which means that $2B(x,x)=0$ , or $B(x,x)=0$ . Therefore, $B$ is an alternating form.

If, however, $\operatorname{char}(K)=2$ , then $B(x,y)=-B(y,x)=B(y,x)$ ; $B$ is a symmetric bilinear form.

If $V$ is finite-dimensional, then every bilinear form on $V$ 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).