  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 used Euler angles 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 \textit{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.
