Let $\mathbb{F}$ be any field of characteristic zero and $SO(n,\mathbb{F})$ denote the group of $n \times n$ rotation matrices in the affine space $\mathbb{A}$ with entries in $\mathbb{F}$. Let $g_i$ be an arbitrary rotation matrix that leaves the $x_i$-axis fixed. For example, \begin{align*} g_1 = \begin{pmatrix} 1 & 0 \\ 0 & h \end{pmatrix} \end{align*} for some $h \in SO(n-1,\mathbb{F})$.
Is it true that for any $n \ge 3$, each $g \in SO(n,\mathbb{F})$ can be written as a product of rotation matrices $g_i$ for $1 \le i \le n$ in whatever permutation where each $i$ occurs only once? It doesn't seem unreasonable (to me) to have this decomposition for a general $n$ but maybe the situation is more complicated over a general $\mathbb{F}$.
Any hints or keywords would be greatly appreciated, as I don't know where to start.
Note: When $n = 3$ and $\mathbb{F} = \mathbb{R}$, I believe this is related to the so-called Euler angles.