We introduce a general setting for multidimensional dispersionless
integrable hierarchy in terms of differential $m$-form $\Omega_m$
with the coefficients satisfying the Pl\"ucker relations, which is
gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients)
are (locally) holomorphic with respect to one of the variables (the spectral variable).
We demonstrate that this form defines a hierarchy
of dispersionless integrable equations
in terms of commuting vector fields locally holomorphic in the spectral variable.
The equations of the hierarchy are given by the gauge-invariant closedness equations.