It could be that the preprint where I found this identity has a typo or that it is simply wrong, but I have been trying to see if this is true:

\begin{equation} \int \left(\nabla\times F_{\bf B}\right)C({\bf x}) \left(\nabla\times G_{\bf B}\right) d{\bf x} =-2\int{\langle \nabla,G_{\bf B}\times F_{\bf B}\rangle d{\bf x}}=0. \end{equation}

where $C({\bf x})$ is a $3\times 3$ matrix given by \begin{equation} C({\bf x}) = \left[ {\begin{array}{ccc} 0 & x_3 & -x_2 \\ -x_3 & 0 &x_1 \\ x_2 & -x_1 & 0 \end{array} } \right] \end{equation}

and $F_{\bf B}$ and $G_{\bf B}$ are smooth vector fields

I cannot prove it. Am I doing something wrong?