  Baryonic matter is studied in the Skyrme model by taking into account the
roles of $\pi,$ $\rho$, and $\omega$ mesons through the hidden local symmetry
up to $\mathcal{O}(p^4)$ terms including the homogeneous Wess-Zumino (hWZ)
terms. Using the master formulas for the low energy constants derived from
holographic QCD models the skyrmion matter properties can be quantitatively
calculated with the input values of the pion decay constant $f_\pi$ and the
vector meson mass $m_\rho^{}$. We find that the hWZ terms are responsible for
the repulsive interactions of the $\omega$ meson. In addition, the
self-consistently included $\mathcal{O}(p^4)$ terms with the hWZ terms is found
to increase the half skyrmion phase transition point above the normal nucleon
density.
