In Marsden "Foundation of Mechanics", it is said that the properties defining the exterior derivative are readily proven to be satisfied by the invariant formula
$d\omega(u)(e_0,...e_k)=\sum_{i=1}^k(-1)^i D\omega(u)·e_i(e_0,...,\hat e_i,...e_k)$
where $e_0,...e_k$ are the basis of the tangent space at point $u$. However, I see no way of proving that this formula satisfies $d(\omega\wedge\rho)=d\omega\wedge\rho + (-1)^k\omega\wedge d\rho$, being $\omega$ a k-form and $\rho$ a m-form, in a coordinate-free way. It seems extremely difficult to prove this property without proving actually that the invariant formula is equal to the coordinate-based definition.
Any clues?
PS: actually, in the book "Manifolds, tensor analysis and applications" by the same author, they don't even try to follow this path and instead prove that both definitions boil down to the coordinate-based one.