Exterior derivative of differential forms as Fréchet derivative?

Question

Let's consider differential forms defined on a Riemannian manifold. Then the tangent bundle may be locally considered as a Banach space and the forms are real valued functions defined in a Banach space. Thus I think we can think of the Fréchet derivative of the differential forms. Does this coincide with the exterior derivative of the differential forms? I'm not so familiar with differential forms and manifolds. I know the rules to use them but am struggling to get an intuitive picture. So the correspondence is just a vague idea I have. I'd like to know whether this picture is plausible or wrong.