Solution
= Solution
Write $\mathcal A=aJ$ with the scalar one-form
$$
a=\langle dx,y\rangle=-\langle x,dy\rangle.
$$
The Lie algebra $\mathfrak o(2)$ is abelian, so $[\mathcal A\wedge\mathcal A]=0$. Moreover
$$
da((u_1,v_1),(u_2,v_2))
=\langle u_2,v_1\rangle-\langle u_1,v_2\rangle.
$$
Consequently
$$
\mathcal F((u_1,v_1),(u_2,v_2))
=\bigl(\langle u_2,v_1\rangle-\langle u_1,v_2\rangle\bigr)
\begin{pmatrix}0&-1\\1&0\end{pmatrix},
$$
as required.