We investigate the random close packing density, $\phi_\textrm{RCP}$, of polydisperse hard sphere systems using a theoretical framework based on the equilibrium model of crowding. We derive a closed-form solution for $\phi_\textrm{RCP}$ in terms of the moments of the diameter distribution, enabling an analytical exploration of the effects of polydispersity ($\delta$) and skewness ($S$) on packing density. For a binary mixture, it is possible to explore a broader range of dependence of $\phi_\textrm{RCP}$ on $\delta$ for a given $S$ or on $S$ for a given $\delta$. We show that the dependencies of $\phi_\textrm{RCP}$ on skewness for a variety of continuous distributions collapse onto a theoretical master curve obtained for the binary mixture case. By correcting the theory so that it obeys known exact limiting behaviours for extreme size asymmetry, our analytical predictions not only agree with previously obtained numerical results, but also predict previously unexplored regions of the $\phi_\textrm{RCP}$ parameter space.