I need to set the rotation and position of an object using a rotation matrix. From the matrix the up vector, the right vector and the forward vector that I desire. What is the best way to turn this into information (a Quaternion I assume) I can use on a unity game object.
You can use Quaternion.Lookrotation to create a rotation from a forward and up vector (the right vector follows automatically).
You cannot set the transformation matrices directly in a transform (their values are read-only), but the LookRotation function should suffice for almost all purposes. If you really need to set the matrices (because you need shearing, for example), you’re only option is to pass your matrix to a shader with a custom vertex shader.