Given two vectors:
$$ v_1=\sigma_x x_1+\sigma_y y_1\\ v_2=\sigma_x x_2 +\sigma_y y_2 $$
The area can be computed as the wedge product
$$ \begin{align} A&=v_1\wedge v_2 \\ &=(\sigma_x x_1 + \sigma_y y_1)\wedge(\sigma_x x_2+\sigma_y y_2)\\ &=(x_1y_2-x_2y_2)\sigma_x\wedge \sigma_y \end{align} $$
My question is can we extend this concept to multivectors, and if so, what is the geometric meaning of the "multi-area"?
$$ u_1=a_1+\sigma_x x_1+\sigma_y y_1\\ u_2=a_2+\sigma_x x_2 +\sigma_y y_2 $$
$$ \begin{align} A_M&=(a_1+\sigma_x x_1+\sigma_y y_1)\wedge (a_2+\sigma_x x_2 +\sigma_y y_2)\\ &=a_1 a_2 + (a_2 x_1 + a_1 x_2) \sigma_x + (a_2 y_1 + a_1 y_2) \sigma_y + (a_2 b_1 + a_1 b_2 - x_2 y_1 + x_1 y_2) \sigma_x \wedge \sigma_y \end{align} $$