The other way around is to define a rotation as the transformation that relates two orthonormal bases. Looking at the two-dimensional case, you would have
$$
\vec e_x = a \vec e'_x + b \vec e'_y \quad \mbox{and} \quad \vec e_y = c \vec e'_x + d \vec e'_y.
$$
From here you can apply the orthonormality conditions of both systems in asserting that ##\vec e_x^2 = \vec e'^2_x = \vec e_y^2 = \vec e'^2_y = 1## and ##\vec e_x \cdot \vec e_y = \vec e'_x \cdot \vec e'_y = 0##. This will give you a set of relations among the coefficients ##a##, ##b##, ##c##, and ##d## (please derive these relations yourself). You should get three independent relations and since you have four parameters, you will be left with one free parameter (the angle of rotation). You will be able to make the interpretation of the angle of rotation by looking at the inner product between ##\vec e_x## and ##\vec e'_x##.