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