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As we know, we can write the cross product between two three dimensional vectors as a matrix-vector product.

Let two vectors are $V = \left[ \begin{array}{c} V_1 \\ V_2 \\ V_3 \end{array} \right] \in \mathbb{R}^{3 \times 1}$,

$V^0 = \left[ \begin{array}{c} V_1^0 \\ V_2^0 \\ V_3^0 \end{array} \right] \in \mathbb{R}^{3 \times 1}$,

Then, we can write the cross product between both vector as matrix-vector product: $V^0 \times V = \left[ \begin{array}{ccc} 0 & -V_3^0 & V_2^0\\ V_3^0 & 0 & -V_1^0\\ -V_2^0 & -V_1^0 & 0 \end{array} \right] . \left[ \begin{array}{c} V_1 \\ V_2 \\ V_3 \end{array} \right]$

However, if we consider general n-dimensional vectors (n = 4-5), could we still write it as dot product between a known vector V0 and unknown vector V?

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  • $\begingroup$ Please clarify what you mean. The cross product in not defined in $\mathbb{R}^n$ for $n \neq 3$. $\endgroup$ – Hans Lundmark Jun 29 '17 at 15:59
  • $\begingroup$ related Is the vector cross product only defined for 3D? $\endgroup$ – Weaam Jun 29 '17 at 16:03
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    $\begingroup$ You can get something similar in $\mathbb{R}^7$, but it doesn't have all the same properties as the cross product. $\endgroup$ – probably_someone Jun 29 '17 at 17:11

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