A rigid body $B$, by definition, is such that, for every point $O\in B$ there is a triple of orthonormal axes ${\bf e}_1, {\bf e}_2, {\bf e}_3$ centered at $O$ such that the material points $Q$ of body $B$ have constant (in time) position vectors when refereed to that frame:
$Q-O = \sum_{k=1}^e x_k(Q) {\bf e}_k$, independently of the motion of $B$.
Now consider another reference frame $O'$, ${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$, where the body appears to move.
If you know the position in thaty space of $O$ and ${\bf e}_1, {\bf e}_2, {\bf e}_3$, you completely know the position of all points of $B$.
To describe the motion of the body in the space of $O'$, ${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$ we can always proceed as follows.
Assign the translatory motion of $O$:
$$O= O' + \sum_{j=1}^{3}x_j(t) {\bf e}'_j\:.$$
Define a third triple of axes ${\bf e}''_1, {\bf e}''_2, {\bf e}''_3$ centered on $O$ and always parallel to ${\bf e}'_1, {\bf e}'_2, {\bf e}'_3$.
Describe how the axes ${\bf e}_1(t), {\bf e}_2(t), {\bf e}_3(t)$ (at rest with $B$) are seen in the triple in 2. This just amounts to assigning a map $R=R(t) \in O(3)$
such that
$${\bf e}_j(t)= \sum_{k=1}^3 R_{jk}(t){\bf e}''_k $$
where every $R(t)$ is a rotation matrix.
It is clear that all the procedure can be performed for every choice of the point $O\in B$. Not only, you can also choose a point $O_1$ which is at rest in the space defined by $O$, ${\bf e}_1, {\bf e}_2, {\bf e}_3$ instead of $O$ itself. It does not matter if $O_1$ is a material point of $B$. It is sufficient that it is at rest with $B$.
