Your equations are insufficient to determine the rotation $T$. To see this, consider a rotation $R$ with axis $\left<v\right>$; if $T$ is any solution then so is $T\circ R$, since compositions of rotations are again rotations. (You can also compose on the right with rotations with axis $\left<T(v)\right>$, although this gives no other alternative solutions than those found by the first method.)
The axis of any rotation that solves this problem must lie in the reflection plane (perpendicular bisector) $H$ between the vectors $v$ and $T(v)$, since any point of the axis obviously stays at equal distances form the two. Moreover any line $l$ in $H$ and passing through the origin can be the axis of a rotation solving the problem. To see this, compose the reflection in $H$ with the unique reflection fixing both $l$ and $T(v)$ (i.e., reflection in the plane spanned by $l$ and $T(v)$). This composition is a rotation, and it sends $v$ to $T(v)$.
For instance, one gets a rotation by a minimal angle by choosing $l$ to be perpendicular to both $v$ and $T(v)$: the line spanned by $(0,1,-1)$. Compute the reflection in $H$ followed by the one in the plane spanned by $(1,0,0)$ and $(0,1,-1)$ (the latter is simple: $(x,y,z)\mapsto(x,-z,y)$) to find a concrete solution.