Definition
A square matrix $A$ is said to be orthogonal if $$ AA^T=I $$ that is a square matrix is orthogonal if their columns determine an orthonormal reference frame.
Now if $\mathscr E:=\{\hat e:i=1,\dots,n\}$ is the standard basis, then the points $$ E_i=O+\hat e_i $$ belong to the $n$-persphere $\Bbb S^n$ and obviously determine an orthonormal affine frame $\mathcal E$: so we can surely imagine to move them rigidly along $\Bbb S^n$ to an another given orthonormal affine frame $\mathcal F$, since the linear transformation whose matrix $Q$ is defined by the orthonormal vectors of $\mathcal F$ does this; but unfortunately I am not able to prove that I can do this continuously, that is I am not able to prove that, for any orthonormalisation affine frame $\mathcal F$, there exists a collection $\mathfrak Q:=\{Q_t :t\in [0,1]\}$ of orthogonal matrices such that $$ Q_0=I\,\,\,\text{and}\,\,\,Q_1=Q $$ and such that the function $$ \varphi(x,t):=Q_tx $$ is continuous on $\Bbb R^n\times[0,1]$ so that I thought to pay a specific question. I point out I am really interested to the three-dimensional case. So, could anyone help me, please?