The linear, second-order, ordinary differential equations governing the free-vibration characteristics, in vacuo, of discretised systems executing, at equilibrium, steady rotational motion about a fixed point may be expressed in the well-known matrix-vector form involving real symmetric and skew-symmetric coefficient matrices. Less well known is the fact that the corresponding Hamiltonian first-order system may be cast into a special form involving a skew-symmetric system matrix. In this paper the computational merits of this special form are exploited in the calculation of the natural frequencies and modes of the rotating system.