We consider the interaction between skyrmions at small separations, and argue that for identical skyrmions it varies as 1/ r, where r is the distance between them. There are significant many-body forces which have the effect of making interaction energies small except for skyrmions in direct contact with one another. Our results differ considerably from those found using the product ansatz, in which the unitary matrix describing the pion field is taken to be a product of the matrices for free skyrmions, and we show how these differences may be resolved. We also show how the n 4 3 behaviour of the energy density of dense matter at high baryon densities n may be understood in terms of the skyrmion-skyrmion interaction.