<s>[INST] What type of product? $\mathbf{a} \circledast \mathbf{b}:= \mathbf{a}\mathbf{b}^T - \mathbf{b}\mathbf{a}^T$ What do you know about the names and properties of the products of the following definitions?
\begin{align}
&\mathbf{a,b}\in\mathbb{R}^N, \mathbf{C}\in\mathbb{R}^{N\times N}\\
&\mathbf{C} = \mathbf{a} \circledast \mathbf{b}:= \mathbf{a}\mathbf{b}^T - \mathbf{b}\mathbf{a}^T
\end{align}
I found some propeties.
\begin{align}
& \mathbf{b} \circledast \mathbf{a} = - \mathbf{a} \circledast \mathbf{b} = (\mathbf{a} \circledast \mathbf{b})^T\\
& \mathbf{a} \circledast (\beta \mathbf{b} + \gamma \mathbf{c}) = \beta (\mathbf{a} \circledast \mathbf{b}) + \gamma (\mathbf{a} \circledast \mathbf{c})
\end{align}
In addition, let $\mathbf{x} = \alpha \mathbf{a} + \beta \mathbf{b}, \mathbf{A} = [\mathbf{a}~\mathbf{b}] \in \mathbb{R}^{N \times 2}$,
\begin{align}
& \mathbf{x}^T (\mathbf{a} \circledast \mathbf{b}) \mathbf{x} = 0 \\
&\|(\mathbf{a} \circledast \mathbf{b}) \mathbf{x}\|_2 = \sqrt{\det \mathbf{A}^T\mathbf{A}} \| \mathbf{x} \|_2 
\end{align}
therefore, $\mathbf{R} = \frac{1}{\sqrt{\det \mathbf{A}^T\mathbf{A}}}\mathbf{a} \circledast \mathbf{b}$ is $\pi/2$ [rad] rotation matrix for $\mathbf{x} \in \mathrm{range}\mathbf{A}$.
Do you know the name of this product?
Can you find any other properties about this product?

We now understand that this product is closely related to the wedge product.
How can I make this product more interesting as wedge product?
For example, the wedge product can be stacked many times.
\begin{align}
a \wedge b \wedge c
\end{align}
But my product cannot be stacked.
 [/INST] This looks like the representation of the wedge product $a\wedge b$ as a skew-symmetric matrix. (Note that $\mathbf C$ is skew-symmetric.) If $a=\sum a_ie_i$ and $b=\sum b_je_j$, then $a\wedge b = \sum\limits_{i<j} (a_ib_j-a_jb_i)e_i\wedge e_j$.
(You might read more about exterior algebra.)
</s>