covariance of cross product of two vectors

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I have two independent vectors in 3D and know the covariance matrix of each. What will be the covariance of cross products of above vectors.

In particular what will be the covariance of cross product if each of vectors covariance matrixes are just a diagonal matrix or even scaled by a scalar identity matrix?

Actually I need to come with covariance matrix for vectors cross product to use it in Kalman filter.