We construct analytic (3 + 1)-dimensional Skyrmions living at finite baryon density in the SU(N) Skyrme model that are not trivial embeddings of SU(2) into SU(N). We use Euler angle decomposition for arbitrary N and the generalized hedgehog ansatz at finite baryon density. The skyrmions of high topological charge that we find represent smooth baryonic layers whose properties can be computed explicitly. In particular, we determine the energy-to-baryon charge ratio for any N showing the smoothness of the large-N limit. The closeness to the BPS bound of these configurations can also be analyzed. The energy density profiles of these finite density skyrmions have a "lasagna-like" shape, in agreement with recent experimental findings. The shear modulus can be precisely estimated as well, and our analytical result is close to recent numerical studies in the literature.