Let $e_1,e_2\in\mathbb{R}^2$ be the standard basis vectors. For $\theta\in\mathbb{R}$, let $R_\theta\in O(2)$ denote the planar rotation determined by $R_\theta e_1=(\cos\theta)e_1+(\sin\theta)e_2$ and $R_\theta e_2=(-\sin\theta)e_1+(\cos\theta)e_2$. Let $R_{e_1}\in O(2)$ denote reflection across the line $e_1^\perp=\operatorname{span}(e_2)$, so that $R_{e_1}e_1=-e_1$ and $R_{e_1}e_2=e_2$.

Then every $A\in O(2)$ has exactly one of the following two forms:

\begin{align*} A=R_\theta \end{align*}

for some $\theta\in\mathbb{R}$, or

\begin{align*} A=R_\theta R_{e_1} \end{align*}

for some $\theta\in\mathbb{R}$. In the first case $\det A=1$, and in the second case $\det A=-1$.