It is important to include many-electron correlation effects in photoionisation calculations for the purpose of explaining experimental observations, for example the one-photon photoionisation cross section. This work presents a method that includes correlation effects within the Random Phase Approximation with Exchange (RPAE) by directly solving a system of linear equations. This is done in the context of computing photoionisation delays for the interpretation of attosecond interferometry schemes like RABBIT. Benchmark results on the one-photon photoionisation cross section, for neon and xenon, are presented. They illustrate the advantages of the method compared to previous, self-consistent, implementations. The new method allows for a detailed scan over photon energies, and can resolve autoionising resonances close to thresholds.