  In this paper we develop algorithms for orthogonal similarity transformations
of skew-symmetric matrices to simpler forms. The first algorithm is similar to
the algorithm for the block antitriangular factorization of symmetric matrices,
but in the case of skew-symmetric matrices, an antitriangular form is always
obtained. Moreover, a simple two-sided permutation of the antitriangular form
transforms the matrix into a multi-arrowhead matrix. In addition, we show that
the block antitriangular form of the skew-Hermitian matrices has the same
structure as the block antitriangular form of the symmetric matrices.
