Angular velocity and Skew Symmetric Matrices

A matrix S is said to be skew symmetric if and only if: S^T + S = 0.
Thus S contains only three independent entries and every 3 × 3 skew symmetric matrix has the form:

If a = (a_x, a_y, a_z)' is a 3-vector, the skew symmetric matrix S(a) can be defined as:

Important properties of the matrix S(a):

• Linearity: for any vector a and b belonging to R^3 and scalars alpha and beta:
• S(alpha*a + beta*b) = alpha*S(a) + beta*S(b)
• Calculation of cross project: for any vector p = (p_x, p_y, p_z)':
• S(a)*p = a.cross(p)
• For an orthogonal matrix (such as rotation) R in SO(3), and a,b are vector in R^3:
• R(a.cross(b)) = (R*a).cross(R*b)
• For R in SO(3) and a vector a belongs to R^3, we have:
• 
• the deviation is:
  

• Computing the derivative of the rotation matrix R is equivalent to a matrix multiplication by a skew symmetric matrix S, that is:
• 

Velocity of a point on a rotating rigid body which is moving with a linear velocity is derived in:

http://en.wikipedia.org/wiki/Angular_velocity#Rigid_body_considerations

Reference:
Skew symmetric matrix