### How to create/use matrices valued in differential forms?

First, for context, I am working with holomorphic Hermitian vector bundles, and I need to compute the connection and curvature matrices, and compute some representatives for Chern classes and the Chern form.

Ultimately I want to have a matrix which is valued in differential froms, and when I multiply matrices, I want the component-wise multiplication to be the wedge product of forms.

When I naively try to build a matrix of forms 'by hand', or when I pre-define my matrix space to be valued in the module of differential forms, I get an error telling me that the space of differential forms is a not a ring.

EDIT: For example,

import ~~sys
~~sys

import ~~mpmath
~~mpmath

sys.modules['sympy.mpmath'] = ~~mpmath
~~mpmath

M = Manifold(2, 'M', ~~field='complex')
~~field='complex')

U = ~~M.open_subset('U')
~~M.open_subset('U')

c_xy.<x,y> = ~~U.chart()
~~U.chart()

e_xy = ~~c_xy.frame()
~~c_xy.frame()

a = M.diff_form(2, ~~name='a')
a.parent()
~~name='a')

a.parent()

eU = ~~c_xy.frame()
a.degree()
~~c_xy.frame()

a.degree()

a[eU,0,1] = x*y^2 + 2*~~x
a.display(eU)
~~x

a.display(eU)

A = matrix([[a, a],[a, a]])

prints out
︡"Module Omega^2(M) of 2-forms on the 2-dimensional complex manifold M\n"}︡{"stdout":"2\n"
"a = (x*y^2 + 2*x) dx/\dy\n"
and tells me that
"TypeError: base_ring (=Module Omega^2(M) of 2-forms on the 2-dimensional complex manifold M) must be a ring"