Skew Symmetric Matrix

A skew-symmetric matrix A is a square matrix whose transpose equals its negative, i,e. $A^T=-A$

$${\mathbf{a}}^{\wedge }=\left[\begin{array}{ccc}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a2& a_1& 0\end{array}\right]$$

The main use case is it can simplify the Cross Product between two matrix into dot product.

$$\begin{array}{c}\vec{a}\times \vec{b}=a^{\wedge }\cdot \vec{b}\end{array}$$

Proof §

Let’s assume $\vec{a}={\left[\begin{array}{ccc}{a}_1& a_2& a_3\end{array}\right]}^T$ and $\vec{b}={\left[\begin{array}{ccc}{b}_1& b_2& b_3\end{array}\right]}^T$

$$\begin{array}{rl}\vec{a}\times \vec{b}& =\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]\\ & =\left[\begin{array}{ccc}0& -a_3& a_2\\ a_3& 0& -a_1\\ -a2& a_1& 0\end{array}\right]\cdot \left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\\ & =a^{\wedge }\cdot \vec{b}\end{array}$$