We study collective excitations in a superfluid neutron gas at zero temperature within the quasiparticle random-phase approximation. The particle-hole residual interaction is obtained from a Skyrme functional, while a separable interaction is used in the pairing channel, which gives a BCS gap that is very similar to the one obtained with a realistic nucleon-nucleon interaction. In accordance with the Goldstone theorem, we find an ungapped collective mode (analogous to the Bogoliubov-Anderson mode). At low momentum, its dispersion relation is approximately linear and its slope coincides with the hydrodynamic speed of sound calculated with the Skyrme equation of state. The response functions are compared with those obtained within the Landau approximation. We also compute the contribution of the collective mode to the specific heat of the neutron gas.