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If I understand you correctly you want an orientation/a new coordinate frame that is perpendicular to n, and "directed with v" (but v is not necessarily perpendicular to n).
The easiest might be to create a rotation matrix. n is one entry. A second one can be derived by n cross v. This should make sure that this entry is perpendicular to n and directed from v. The third one should be the cross of the other two.
My guess is that n cross v should go to column 1 and n to column 2, but not guarantees.
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