Let $\mathbb{F}_p^3$ be a $3$-dimensional vector space over $\mathbb{F}_p$ with $p$ odd. For any $\mathbf{x}\in \mathbb{F}_p^3$ define its "norm" $\lVert \mathbf{x}\rVert=x_1^2+x_2^2+x_3^2,$ where $\mathbf{x}=(x_1,x_2,x_3)$.
Remark: If $\mathbf{u},\mathbf{v}\in \mathbb{F}_p^3$ such that $\lVert \mathbf{u}\rVert=\lVert \mathbf{v}\rVert\neq 0$, then there exists $g\in SO_3(\mathbb{F}_p)$ such that $\mathbf{u}=g\mathbf{v}$. We note that $g$ is not unique, but we can achieve uniqueness by considering the quotient $O_3(\mathbb{F}_p)/O_2(\mathbb{F}_p)$, i.e. there is a unique $O_2(\mathbb{F}_p)$-coset of elements $g$ with the property.
The last couple days I was trying to prove that remark but I failed. I was trying to construct such orthogonal matrix with determinant $1$ by hand but it did not work out. Can anyone provide the proof of that remark please?
Many thanks!