Given the Cartesian vectors $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$ and some arbitrary vector in 3D-space $\mathbf{t}$, is it possible to define a unique mapping (matrix operator) that maps $\mathbf{e}_2$ to some $\mathbf{m}$, and some $\mathbf{e}_3$ to $\mathbf{n}$, such that $\mathbf{m} \perp \mathbf{n} \perp\mathbf{t}$? ($\mathbf{m}$ and $\mathbf{n}$ are not given)
In other words, given some $\mathbf{t} \in \mathbb{R}^3$, find $\boldsymbol{Q}\in \mathbb{R}^{3 \times 3}$ such that $$ \boldsymbol{Q}:\{ \mathbf{e}_2, \mathbf{e}_3\} \to \{ \mathbf{m}, \mathbf{n}\} \, ,\quad \text{s.t.}\quad \mathbf{m} \perp \mathbf{n} \perp\mathbf{t} $$
Is this possible to do with a single formula?, or is the only solution algortihmic in nature (test the cross product against two possible candidates, at least one won't be parallel to $\mathbf{t}$)?