One more example: for $d=8$ based on octonions. And this are all possible dimensions. $R_1$ takes vectors of length one to vectors of length one hence it is an isometry. By multiplying from the left by $R_1^{-1}$ we may assume that $R_1=\mathop{\rm Id}$. Then for every $v\in S^{d-1}$, the unit sphere in $\mathbb{R}^d$, the map $v\mapsto(R_2v,R_3v,\ldots,R_dv)$ gives a parallelization of the tangent bundle of $S^{d-1}$. Then by a celebrated Bott-Kervaire-Milnor theorem such a parallelization exists only when $d\in\{1,2,4,8\}$.

One more example: for $d=8$ based on octonions. And this are all possible dimensions. $R_1$ takes vectors of length one to vectors of length one hence it is an isometry. By multiplying from the left by $R_1^{-1}$ we may assume that $R_1=\mathop{\rm Id}$. Then for every $v\in S^{d-1}$, the unit sphere in $\mathbb{R}^d$, the map $v\mapsto(R_2v,R_3v,\ldots,R_dv)$ gives a parallelization of the tangent bundle of $S^{d-1}$. Then by a celebrated Bott-Kervaire-Milnor theorem such a parallelization exists only when $d\in\{1,2,4,8\}$.

Edit: More explicitly, the solution for $d = 8$ is given by $$ \sum_{i=1}^8 a_i R_i = \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 & a_6 & a_7 & a_8 \\ -a_2 & a_1 & -a_4 & a_3 & -a_6 & a_5 & a_8 & -a_7 \\ -a_3 & a_4 & a_1 & -a_2 & -a_7 & -a_8 & a_5 & a_6 \\ -a_4 & -a_3 & a_2 & a_1 & -a_8 & a_7 & -a_6 & a_5 \\ -a_5 & a_6 & a_7 & a_8 & a_1 & -a_2 & -a_3 & -a_4 \\ -a_6 & -a_5 & a_8 & -a_7 & a_2 & a_1 & a_4 & -a_3 \\ -a_7 & -a_8 & -a_5 & a_6 & a_3 & -a_4 & a_1 & a_2 \\ -a_8 & a_7 & -a_6 & -a_5 & a_4 & a_3 & -a_2 & a_1 \\ \end{pmatrix} $$