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We study the properties of hot and dense neutron star matter under the presence of strong magnetic fields using two Skyrme interactions, namely the LNS and the BSk21 ones. Asking for $\beta$--stability and charge neutrality, we construct the equation of state of the system and analyze its composition for a range of densities, temperatures and magnetic field intensities of interest for the study of supernova and proto-neutron star matter, with a particular interest on the degree of spin-polarization of the different components. The results show that system configurations with larger fractions of spin up protons and spin down neutrons and electrons are energetically favored over those with larger fractions of spin down protons and spin up neutrons and electrons. The effective mass of neutrons and protons is found to be in general larger for the more abundant of their spin projection component, respectively, spin down neutrons and spin up protons. The effect of the magnetic field on the Helmhotz total free energy density, pressure and isothermal compressibility of the system is almost negligible for all the values of the magnetic field considered., Comment: 10 pages, 8 figures, 2 tables