Let $\chi(\mathbb{R^3})$ denote the vector space of all smooth vector fields on $\mathbb{R^3}$, and let $A$ be the subspace of $\chi(\mathbb{R^3})$ spanned by $\{X,Y,Z \}$ where
\begin{align*} X &= y {\partial \over \partial z}-z {\partial \over \partial y},\\ Y &= z {\partial \over \partial x}-x {\partial \over \partial z},\\ Z &= x {\partial \over \partial y}-y {\partial \over \partial x}. \end{align*}
The question first ask me to compute the Lie brackets $[X,Y]$,$[Y,Z]$ and $[Z,X]$. Which I did and they are, \begin{align*} [X,Y] &= y {\partial \over \partial x}-x {\partial \over \partial y},\\ [Y,Z] &= z {\partial \over \partial y}-y {\partial \over \partial z},\\ [Z,X] &= x {\partial \over \partial z}-z {\partial \over \partial x}. \end{align*} Then how can I deduce that there is an isomorphism from $\mathbb{R^3}$ to $A$ so that the cross product in $\mathbb{R^3}$ corresponds to the Lie bracket of vector fields.
You have just shown that $$[X,Y]=-Z,\qquad [Y,Z]=-X,\qquad [Z,X]=-Y.$$ Now it remains to compare this with $$ \mathbf{e}_x\wedge \mathbf{e}_y =\mathbf{e}_z,\qquad \mathbf{e}_y\wedge \mathbf{e}_z =\mathbf{e}_x,\qquad \mathbf{e}_z\wedge \mathbf{e}_x =\mathbf{e}_y.$$