Starting with Cartan's magic formula as well as the formula $$ (\mathcal{L}_Y \omega)(X_1, ..., X_k) = Y(\omega(X_1, ... , X_k)) - \omega([Y,X_1], X_2, ..., X_k) - \cdots - \omega(X_1, ..., [Y,X_k]) $$
I am trying to prove the following coordinate-free formula for the exterior derivative (using the formulas above, not coordinates, which makes this different than other questions about the different definitions for exterior derivative on this site):
$$d\omega(X_0, ... , X_k) = \sum_i (-1)^i X_i(\omega(X_0, ... , \hat{X_i}, ... , X_k)) + \sum_{i < j}(-1)^{i+j}\omega([X_i, X_j], X_0, ... , \hat{X_i}, ... , \hat{X_j}, ..., X_k)$$
I am familiar with the coordinate-based proof in Lee that this is equivalent to the axiomatically defined exterior derivative defined (showing first that this expression is $C^{\infty}(M)$-linear and then showing equivalence of this formula and the usual coordinate expression of exterior derivative in a coordinate patch). However, I am not sure how to use Cartan's magic formula and the expression above to obtain the coordinate free expression for d shown above.
Here's how it goes for a $1$-form $\omega$. To do it for $k\ge 2$ just requires more bookkeeping.
You're starting with \begin{align*} (\mathscr L_X\omega)(Y) &= X(\omega(Y)) - \omega(\mathscr L_XY) \\ \iota_X d\omega &= \mathscr L_X\omega - d(\iota_X\omega). \end{align*} Beginning with the latter, we have \begin{align*} d\omega(X,Y) &= (\mathscr L_X\omega)(Y) - d(\omega(X))(Y) = X(\omega(Y)) - \omega(\mathscr L_XY) - Y(\omega(X)) \\ &= X(\omega(Y)) - Y(\omega(X)) - \omega([X,Y]), \end{align*} as desired.