n - forms as Linear Transformations
\[dx_1 \wedge dx_2 \wedge ... \wedge dx_n\]
is n - form operating on vectors \[(\mathbf{v}_1 , \mathbf{v}_2 , ..., \mathbf{v}_n )\]
then\[\begin{equation} \begin{aligned} & dx_1 \wedge dx_2 \wedge ... \wedge dx_n (k_1 (\mathbf{v}_1 , \mathbf{w}_2 , ..., \mathbf{v}_n ) + k_2 (\mathbf{w}_1 , \mathbf{w}_2 , ..., \mathbf{w}_n )) \\&= k_1 dx_1 \wedge dx_2 \wedge ... \wedge dx (k_1 (\mathbf{v}_1 , \mathbf{w}_2 , ..., \mathbf{v}_n ) \\ &+ k_2 dx_1 \wedge dx_2 \wedge ... \wedge dx_n (\mathbf{w}_1 , \mathbf{w}_2 , ..., \mathbf{w}_n )) \end{aligned} \end{equation} \]
This means that we can represent an n - form by a matrix.
The set of n forms is also linear for each value of n, so that if
\[L_1\]
and \[L_2\]
are n - forms then so is \[k_1 L_1 + k_2 L_2\]
where \[k_1 , \: k_2\]
are scalars. 
