I'm doing some calculations with qutrits and I need a unitary matrix $U$ that does the following:
$$U|00\rangle = |12 \rangle - | 21\rangle $$
$$U|11\rangle = |20 \rangle - | 02\rangle $$
$$U|22\rangle = |01 \rangle - | 10\rangle $$
where $|0\rangle$,$|1\rangle$, $|2\rangle$ are the basis states. I don't care how it acts on the other states as long as it acts in the way prescribed earlier. Until now I've been using ad-hoc methods for finding such unitaries. In summary, I require two things: the unitary that fulfills the conditions mentioned above and if possible some general method to find unitaries that perform certain transformations over qutrits or qudits.