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I obtain a 3×3 matrix corresponding to a general rotation in 3D space via some program. However, let's say that for some reason the third row in the matrix is damaged and filled with zeroes, which of course makes the matrix incorrect. I simply cannot fix the process obtaining this matrix, therefore I need to reconstruct the third row based on the first two rows and the fact that it is a rotation. Is it possible? If so, what is the formula for the values?

If not, I am also able to obtain the Euler angle around the Z axis (yaw) for this exact same rotation. Would it then be possible to reconstruct the third row?

I suppose a description of the Euler rotation system and the rotation order may be required for determining the correct formula; please see this question and mainly the answer for the description.

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    The third row will be the [cross product](https://en.wikipedia.org/wiki/Cross_product) of the first two rows. – 2017-01-05
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    The 3 row vectors (or 3 column vectors) of a rotation matrix are orthonormal among themselves. if you know two of them, then up to a sign, the third one is the cross product. – 2017-01-05
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    @achillehui: The sign is determined by the necessity of a rotation matrix to have determinant $+1$. – 2017-01-06
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    See also answers from http://math.stackexchange.com/questions/2046270/is-5-entries-for-an-orthogonal-matrix-enough-for-reconstruction for clarification. – 2017-01-07

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