I have a number of questions/comments:
1. How are you thinking of "an XYZ rotation"? Do you mean rotations about each axis? Or do you mean that you're simply inputting the x, y, and z components?
2. If the latter is the answer to my question # 1 above, then here's another question: the xyz coordinate system IS an orthonormal basis. Is that not adequate?
3. If you need to translate from one orthonormal basis to another orthonormal basis, then you're dealing with a rotation. You need to know what that rotation does to the xyz basis. Then you'll know what it does to any other vector.