An exercise of my last year's linear algebra class asks as follows:
Determine the type of the quadratic form $q: \bigwedge^2 \mathbb{R}^4 \to \bigwedge^4 \mathbb{R}^4, x \mapsto x \wedge x$.
Question 1: We have only introduced quadratic forms of bilinear forms and I can't recall our professor talking about different "types" of quadratic forms. What types of quadratic forms are there?
Question 2: In the standard solution, one is adviced to proceed as follows: We have $\dim(\bigwedge^2 \mathbb{R}^4) = 6$ and thus the quadratic form will be represented by a 6x6-matrix no matter what basis we choose. We now choose the basis
$\mathcal{B} = (e_1 \wedge e_2, e_1 \wedge e_3, e_1 \wedge e_4, e_2 \wedge e_3, e_2 \wedge e_4, e_3 \wedge e_4)$
and by plugging $\mathcal{B}$ into the quadratic form, we arrive at the matrix
$M_\mathcal{B}(q) = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}.$
I don't understand how we arrive at this matrix. Can anyone explain?