For a scalar field $ \phi $, we have the "coordinate-free" definition in terms of the exterior derivative: $$ \nabla \phi = \ (\mathrm d \phi)^\flat := \ \langle \mathrm d \phi, \cdot \rangle $$
However, in trying to derive some vector calculus identities, I came into a bit of trouble in finding an expression for the gradient of a vector field $A$. I tried $ (\mathrm dA^{\sharp})^\flat $ but this doesn't seem to work out when you do a simple example in coordinates.
Is there a good simple formulation of this somewhere?
Thanks in advance.