# Skew symmetric matrix of vector follow up - How to obtain skew matrix for N-dimensional vector

I am trying to understand usage of skew matrices:

1) It is my understanding from an earlier question in this forum that for vectors A and B, that A cross B is the same as the "skew symmetric matrix" of A times B. Is this correct?

2) If the above is correct, I did a simple example in matlab where A = [1 0 0] and B = [0 1 0]. The obvious cross product is C = [0 0 1]; The skew multiplication is Sij = AiBj - AjBi which results in [0 1 0;-1 0 0; 0 0 0], but the skew of C, or AxB, is [0 -1 0; 1 0 0; 0 0 0]. Looks like the two results are negated or transposed. So I am not sure how this is equivalent, but maybe I just don't understand the application. Any explanation is appreciated.

3) If skew multiplication is a generalization to N dimensional vectors of the cross product when N = 3, then is there an example of finding an orthogonal vector to two vectors in 4-D? All the examples I can find show the equivalent skew matrix for a 3-D vector and how that multiplied with another 3-D vector results in the cross product vector, but I see no reference for getting the skew mapping for N dimensional vectors so that you can multiply that with another N-dimensional vector to obtain an orthogonal N-D vector.

Thanks in advance

• Welcome to Math.SE! Please use MathJax to format your posts. – csch2 Apr 8 at 20:02