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Provided that a cohomological model for $$G$$ is known, we describe a method for constructing a basis for $$n$$ -cocycles over $$G$$ , from which the whole set of $$n$$ -dimensional $$n$$ -cocyclic matrices over $$G$$ may be straightforwardly calculated. Focusing in the case $$n=2$$ (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative $$2$$ -cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When $$n>2$$ , this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for $$n=2,3$$ . In particular, we give some examples of improper 3-dimensional $$3$$ -cocyclic Hadamard matrices. [ABSTRACT FROM AUTHOR]