  We study differential $p$-forms on non-smooth and possibly fractal metric
measure spaces, endowed with a local Dirichlet form. Using this local Dirichlet
form, we prove a result on the localization of antisymmetric functions of $p+1$
variables on diagonal neighborhoods to differential $p$-forms. This result
generalizes both the well-known classical localization on smooth Riemannian
manifolds and the well-known semigroup approximation for quadratic forms. We
observe that a related localization map taking functions into forms is
well-defined and induces a chain map from a differential complex of
Kolmogorov-Alexander-Spanier type onto a differential complex of deRham type.
