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 [ (underline{underline A},underline{underline B})oplus (A_1,B_1)oplusdotsoplus(A_t,B_t) ] that is congruent to $(A,B)$, in which $(underline{underline A},underline{underline B})$ is a pair of nonsingular matrices and $(A_1,B_1),$ $dots,$ $(A_t,B_t)$ 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.