When composing n - forms remeber that
\[dx_i \wedge dx_i \wedge ... \wedge dx_n =0\]
and \[\wedge dx_1 \wedge ... \wedge dx_i \wedge dx_j = -\wedge dx_1 \wedge ... \wedge dx_j \wedge dx_i \]
.Example
\[\begin{equation} \begin{aligned} (dx_1 \wedge dx_2) \wedge (dx_1 \wedge dx_2) &= dx_1 \wedge dx_2 \wedge dx_1 \wedge dx_2 \\ &= dx_1 \wedge dx_1 \wedge dx_2 \wedge dx_2=0 \end{aligned} \end{equation}\]
Example
\[\begin{equation} \begin{aligned} (dx_1 + dx_2) \wedge (dx_1 - dx_2) &= dx_1 \wedge dx_1 - dx_1 \wedge dx_2 +dx_2 \wedge dx_1 - -dx_2 \wedge dx_2 \\ &=0 - dx_1 \wedge dx_2 - dx_1 \wedge dx_2 +0 \\ &= -2 dx_1 \wedge dx_2 \end{aligned} \end{equation}\]
Exampl
\[\begin{equation} \begin{aligned} (dx_1 + dx_2) \wedge (dx_1 - dx_2 + dx_3) &= dx_1 \wedge dx_1 - dx_1 \wedge dx_2 +dx_1 \wedge dx_3 \\ &+dx_2 \wedge dx_1 - dx_2 \wedge dx_2 +dx_2 \wedge dx_3 \\ &=0 - dx_1 \wedge dx_2 + dx_1 \wedge dx_3 \\ &- dx_1 \wedge dx_2 -0 + dx_2 \wedge dx_3 \\ &= -2 dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_2 \wedge dx_3 \end{aligned} \end{equation}\]