coordinate tensor with respect to the standard basis vectors

Question

Consider the bilinear form $\phi: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $$\phi(v,w)=\langle(1,0,0)^T,v \times w\rangle$$ Then what is the coordinate tensor with respect to the standard basic vectors?

Answer 1

In the standard basis, $\phi=v_yw_z-v_zw_y=v_iT_{ij}w_j$ (see $r=p_iT_{ij}q_j$ here) with$$T=\left(\begin{array}{ccc}0 & 0 & 0\\0 & 0 & 1\\0 & -1 & 0\end{array}\right).$$