how to calculate the cross product of two multidimensional matrices

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Is the cross product of two 3x3 matrices even defined? Certainly for the dot product we say that the dot product of two 3x1 vectors is a scalar but the dot product of two 3x3 matrices is a 3x3 matrix. Since the cross product of two 3x1 vectors is a 3x1 vector, would we by analogy say that the cross product of two 3x3 matrices would be a 3x3x3 matrix? And would we calculate the third dimension as the vector product of the two 3x1 vectors where they intersect?