Skew symmetric logarithms and geodesics on $O_n(\RR)$
  We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric. 
Alberto Dolcetti, Donato Pertici
math.DG