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If $a \wedge b = a \otimes b - b \otimes a$ then what is $a \wedge b \wedge c$

I know it's supposed to be a trivector but what is it in Matrix form?

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  • $\begingroup$ Anyone have any ideas? $\endgroup$
    – R. Emery
    Nov 10, 2017 at 21:31
  • $\begingroup$ I'm confused. Isn't $a\wedge b=b\otimes a - a\otimes b=\begin{bmatrix}0&a_yb_x-a_xb_y\\a_xb_y-a_yb_x&0\end{bmatrix}$? $\endgroup$ Jul 1, 2022 at 13:42

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$e_1 \wedge e_2 \wedge e_3$ = $ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \\ \end{bmatrix}$ $ \begin{bmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{bmatrix}$ $ \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$

$ a \wedge b \wedge c = a \otimes b \otimes c - a \otimes c \otimes b + c \otimes a \otimes b - c \otimes b \otimes a + b \otimes c \otimes a - b \otimes a \otimes c$

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