If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to know/calculate the orientation of an object. Why do you need to be able to transform those vectors in order to know the orientation?
I initially thought that this rotation matrix was an absolute representation of your orientation, but appearantly it's not.
example of an algorithm using this principle: https://en.wikipedia.org/wiki/Triad_method
Best Answer
In order to know the orientation of an object, you need to first privilege a basis to examine the object with respect to. You need some kind of axes to use as a reference point. If you calculate the rotation matrix with respect to am that basis, it allows you to calculate the orientation of the object.
It should t surprise you that knowing orientation and knowing rotation matrices are the same. After all, the orientation is a measure of an amount of rotation. The rotation matrix you find above is the one that transforms from the basis to the orientation, which is how we specify an orientation.