I've looked upon the following exterior product
$$\sum_r \alpha_{i_1}...\alpha_{i_r}(X^{i_1}dt+H_t^*dx^{i_1})\wedge...\wedge(X^{i_r}dt+H_t^*dx^{i_r})$$
which after some manipulation has to be equal to
$$\sum_r \alpha_{i_1}...\alpha_{i_r}dt \wedge H_t^*(\sum_{j=1}^r (-1)^{j-1} X^{i_j} dx^{i_1}\wedge...\wedge dx^{i_j} \wedge...\wedge dx^{i_r})$$
Through the techniques I've self-taught I wasn't able to get this result. Any help?