Find the matrix $\mathbf{A}$ such that $\mathbf{A}\mathbf{w} = \begin{pmatrix}4 \\ 7 \\ -1\end{pmatrix} \times \mathbf{w}$ for all vectors $\mathbf{w}.$
This is the image in case it's easier to understand:
All I know is that $\mathbf{A}$ is a 3 by 3 matrix. I haven't attempted it because I don't understand what this question is saying. Any advice/help?
Hint: Since it is given for all vectors $w$, so start with $w=e_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$. Then you have $Ae_1=v \times e_1$, where $v$ is the vector given on the right side of the equation. This will give you the first column of $A$.
Note: $v \times e_1$ is very simple to compute if you know what cross-product means.