  In this paper the SU(N) Einstein-Skyrme system is considered. We express the
chiral field (which is not a simple embedding of the SU(2) one) in terms of
harmonic maps. In this way, SU(N) spherical symmetric equations can be obtained
easily for any $N$ and the gravitating skyrmion solutions of these equations
can be studied. In particular, the SU(3) case is considered in detail and three
different types of gravitating skyrmions with topological charge 4, 2 and 0,
respectively, are constructed numerically. Note that the configurations with
topological charge 0 correspond to mixtures of skyrmions and antiskyrmions.
