Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition[2]: p. 38 

This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.

This defines a form with desirable properties for vector spaces over fields of characteristic not equal to 2, but in a vector space over a field of characteristic 2, the definition is equivalent to that of a symmetric form, as every element is its own additive inverse.

This is equivalent to a skew-symmetric form when the field is not of characteristic 2, as seen from

It is easy to check that the commutator of two skew-symmetric matrices is again skew-symmetric: