According to Gallot-Hulin-Lafontaine one has $$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_1,\cdots,X_{i-1},X_0,X_{i+1},\cdots,X_q)$$
It seems to me that it should be $$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_0,\cdots,\hat{X_i},\cdots,X_q)$$
Is this right ?
The correct formula is given on Wikipedia. If the vector fields commute (for example, if the $X_k$'s are the vector fields associated to a coordinate system), then it reduces to your formula.
It's not even clear to me how to interpret the terms for $i=0$ or $i=1$ in their formula, and in any case the factor $(-1)^i$ looks strange, since they would get the alternating signs from the moving of the argument $X_0$ anyway.