Inspired by the low-density Lee-Yang expansion for the energy of a dilute Fermi gas of density $\rho$ and momentum $k_F$, we introduce here a Skyrme-type functional that contains only $s$-wave terms and provides, at the mean-field level, (i) a satisfactory equation of state for neutron matter from extremely low densities up to densities close to the equilibrium point, and (ii) a good-quality equation of state for symmetric matter at density scales around the saturation point. This is achieved by using a density-dependent neutron-neutron scattering length $a\left(\rho \right)$ which satisfies the low-density limit (for Fermi momenta going to zero) and has a density dependence tuned in such a way that the low-density constraint $|a\left(\rho \right)k_F|\le 1$ is satisfied at all density scales.