Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum(A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which (A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A,B) under congruence over an algebraically closed field of characteristic not 2.