Suppose we have a 2 form
\[2dx_1 \wedge dx_2 \]
and vectors \[\mathbf{a}=(1,0,1), \: \mathbf{b}=(-2,3,1)\]
.We want to find the value of the 2 - form at
\[(\mathbf{a}, \mathbf{b})\]
.\[\begin{equation} \begin{aligned} dx_1 \wedge dx_2 (\mathbf{a}, \mathbf{b}) &= det \left( \begin{array}{cc} a_1 & b_1 \\ a_2 & b_2 \end{array} \right) \\ &- det \left( \begin{array}{cc} dx_1(\mathbf{a}) & dx_1(\mathbf{b}) \\ dx_2(\mathbf{a}) & dx_2(\mathbf{b}) \end{array} \right) \\ &= det \left( \begin{array}{cc} 1 & -2 \\ -0 & 3 \end{array} \right) =3 \end{aligned} \end{equation}\]
Hence
\[2dx_1 \wedge dx_2 (\mathbf{a}, \mathbf{b}) =2 \times 3=6\]
.