We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a ${\Bbb Z}^d $ action. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when $d \geq 2$. We propose a definition of the entropy for random group actions which agrees with the classical definition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This definition is based on the fibre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases.