The cross product of two vectors in ${\mathbb R}^3$ is defined by $$\left\lbrack \begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array} \right\rbrack \times \left\lbrack \begin{array}{c} b_1 \\ b_2 \\ b_3 \end{array} \right\rbrack = \left\lbrack \begin{array}{c} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{array} \right\rbrack.$$ Let $\vec{v}= \left[\begin{array}{c} -6\cr 6\cr -7 \end{array}\right].$ Find the matrix $A$ of the linear transformation from ${\mathbb R}^3$ to ${\mathbb R}^3$ given by $T(\vec{x}) = \vec{v} \times \vec{x}.$

$A=$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$