Angular velocity tensor

It is defined as this matrix,

${\displaystyle {\boldsymbol {\omega }}(t)\times A(t)\mathbf {r} _{0}={\begin{pmatrix}0&-\omega _{z}(t)&\omega _{y}(t)\\\omega _{z}(t)&0&-\omega _{x}(t)\\-\omega _{y}(t)&\omega _{x}(t)&0\\\end{pmatrix}}A(t)\mathbf {r} _{0}.}$

It has the important property that when the cross product

${\displaystyle {\boldsymbol {\omega }}(t)\times A(t)\mathbf {r} _{0}}$

is written with the matrix multiplication (A is a orientation matrix), this matrix is a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements,