A formula is derived for a rotation matrix which reduces orthogonally a \(4\times 4\) real skew-symmetric matrix \(A\) to real Schur form. A Jacobi type method is used. By applying the \(4\times 4\) matrices the off- diagonal part of the matrix is annihilated. An essential ingredient of the formula is the parametrization of \(4\times 4\) rotations by pairs of quaternions. For distinct eigenvalues of \(A\) (the eigenvalues are \(\lambda_ 1 i,-\lambda_ 1 i,\dots,\lambda_ n i,-\lambda_ n i\), where \(0\leqq \lambda_ 1<\lambda_ 2<\dots<\lambda_ n\)) the convergence of a special cyclic method is asymptotically quadratic. The resulting algorithm may be attractive in the context of parallel computing.