Cross product between two n-dimensional vector $(n = 4-5)$

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?

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