We use properties of doubly magic nuclei and ab initio calculations of low-density neutron matter to constrain Skyrme equations of state for neutron-rich conditions. All of these properties are consistent with a Skyrme functional form and a neutron-matter equation of state that depends on three parameters. With a reasonable range for the neutron-matter effective mass, the values of the two other Skyrme parameters are well constrained. This leads to predictions for other quantities. The neutron skins for ${}^{208}\mathrm{Pb}$ and ${}^{48}\mathrm{Ca}$ are predicted to be 0.182(12) and 0.173(5) fm, respectively. Other results including the dipole polarizability are discussed.