R = [e1; e2; e3]
is the rotation matrix already, when we assume, that these are the normalized orthogonal vectors of the local coordinate system. To convert between the two reference systems all you need is R and R.' (as long as the translation is ignored).
A vector v=[x;y;z] in the global reference system is
R * v
in the local system. Then you can convert it back to the global system by:
R.' * R * v
and because this must reply v again you get the identity:
R.' * R == eye(3)
such that
R.' == inv(R) % except for rounding errors
You can split this rotation matrix to 3 matrices around specific axes to get a set of Euler or Cardan angles.
You will find many discussions in the net, which end in flamewars, because the users cannot decide if R or R' is the actual rotation, because sometimes the rotation of the vector is meant, and sometimes the rotation of the reference system. Each field of science has its own preferences in this point. See https://www.mathworks.com/matlabcentral/answers/313150-why-is-the-result-of-quaternion-rotation-an-matrix-multiplication-not-the-same
Maybe you are looking for the "helical axes", which can define the relative attitude between two coordinate systems by a vector of translation and a rotation around this vector. If so, please clarify this.
