We study linear, time-invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form. We address several cases, in an increasing order of complexity, and show how dissipative systems can be parametrised. In this process, we also establish a number of results for polynomial matrices that are of independent interest. The present article is a generalisation of our earlier work that dealt with single-input single-output dissipative systems (Pendharkar, I., and Pillai, H.K. (2004), 'A Parametrization for Dissipative Behaviours', Systems and Control Letters, 51, 123-132).