We construct an isomorphism between the $\overline{\partial}$-cohomology
and the cohomology of the fundamental groups of toroidal groups,
and get a standard form of $p$-cocycles, which was given by
Vogt [6] in case of $1$-cocycles. Using differential forms
via the above isomorphism enables us to obtain new results
in higher dimensional cases. An explicit isomorphism between
the Čech cohomology and the cohomology of the fundamental
groups of complex tori is given in [5] (p.14). Our results
give a generalization of this isomorphism to toroidal groups.