Matrix cross product

Question

I'm having a bit of a hard time with cross products and this question in particular. How do we approach these?

Assume that $ \vec{a} \in {R}^3$ \ $\vec{0}$.

Find matrix $A$ , such that, for every $\vec{v} \in R^3 $ the following $A\vec{v} = \vec{a} × \vec{v}.$ Holds true.

Answer 1

Guide:

This is a question in $\mathbb{R}^3$.

Express $a \times v$ explicitly in terms of $v_1, v_2, v_3, a_1, a_2, a_3$ where $a_i$ and $v_i$ are the $i$-th component of $a$ and $v$ respectively.

Answer 2

$$\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \times \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = \begin{pmatrix} 0&-a_3&a_2 \\ a_3&0&-a_1 \\ -a_2&a_1&0 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$$ Now, how did I get to that matrix? Does the matrix representation of a linear transformation sound to you?