We combine the theory of Pommaret bases with a (slight generalisation of a) recent construction by Sköldberg based on discrete Morse theory. This combination allows us the explicit determination of a (generally non-minimal) free resolution for a graded polynomial module with the computation of only one Pommaret basis. If only the Betti numbers are needed, one can considerably simplify the computations by determining only the constant part of the differential. For the special case of a quasi-stable monomial ideal, we show that the induced resolution is a mapping cone resolution. We present an implementation within the CoCoALib and test it with some common benchmark ideals.