Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations

The homogeneous system of matrix equations (XTA+AX,XTB+BX)=(0,0), where (A,B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A,B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.