A rational criterion for congruence of square matrices

With a square complex matrix $A$ we associate the matrix pair consisting of its symmetric part $S(A) = (A + A^T)/2$ and its skew-symmetric part $K(A) = (A - A^T)/2$. We show that square matrices $A$ and $B$ are congruent if and only if the associated pairs $(S(A),K(A))$ and $(S(B),K(B))$ are (strictly) equivalent. This criterion can be verified by a finite rational calculation if the entries of $A$ and $B$ are rational or rational Gaussian numbers.