<P> Three - by - three skew - symmetric matrices can be used to represent cross products as matrix multiplications . Consider vectors a = (a 1 a 2 a 3) T (\ displaystyle \ mathbf (a) = (a_ (1) \ a_ (2) \ a_ (3)) ^ (\ mathrm (T))) and b = (b 1 b 2 b 3) T (\ displaystyle \ mathbf (b) = (b_ (1) \ b_ (2) \ b_ (3)) ^ (\ mathrm (T))). Then, defining matrix: </P> <Dl> <Dd> (a) × = (0 − a 3 a 2 a 3 0 − a 1 − a 2 a 1 0) (\ displaystyle (\ mathbf (a)) _ (\ times) = (\ begin (bmatrix) \, \, 0& \! - a_ (3) & \, \, \, a_ (2) \ \ \, \, \, a_ (3) &0& \! - a_ (1) \ \ \! - a_ (2) & \, \, a_ (1) & \, \, 0 \ end (bmatrix))) </Dd> </Dl> <Dd> (a) × = (0 − a 3 a 2 a 3 0 − a 1 − a 2 a 1 0) (\ displaystyle (\ mathbf (a)) _ (\ times) = (\ begin (bmatrix) \, \, 0& \! - a_ (3) & \, \, \, a_ (2) \ \ \, \, \, a_ (3) &0& \! - a_ (1) \ \ \! - a_ (2) & \, \, a_ (1) & \, \, 0 \ end (bmatrix))) </Dd> <P> the cross product can be written as </P>

A skew symmetric determinant of even order is