v = [ v 1 v 2 v 3 ] {\displaystyle {\mathbf {v}}={\begin{bmatrix}v_{1}\\v_{2}\\v_{3}\\\end{bmatrix}}} .
T ( v ) = v × [ 1 0 0 ] = | i j k v 1 v 2 v 3 1 0 0 | = [ 0 v 3 − v 2 ] {\displaystyle T({\mathbf {v}})={\mathbf {v}}\times {\begin{bmatrix}1\\0\\0\\\end{bmatrix}}={\begin{vmatrix}i&j&k\\v_{1}&v_{2}&v_{3}\\1&0&0\end{vmatrix}}={\begin{bmatrix}0\\v_{3}\\-v_{2}\\\end{bmatrix}}} .
This implies that
That is, T {\displaystyle T} maps identity matrix to [ 0 0 0 0 0 1 0 − 1 0 ] {\displaystyle {\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}}} . Thus, the matrix representation of T {\displaystyle T} is
T = [ 0 0 0 0 0 1 0 − 1 0 ] {\displaystyle \color {blue}T={\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\end{bmatrix}}} .
