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 ?