The convex hull of SO(2) is a flat disc

Idea. Under the linear isomorphism $A\mapsto(z,w)$, $SO(2)$ becomes the unit circle $\{(e^{i\alpha},0)\}$ in the plane $\{w=0\}$, and the convex hull of a circle is the closed disc.

Step 1: $SO(2)$ in coordinates. $R_\alpha x=e^{i\alpha}x$ for $x\in\mathbb{C}$. From the entries: $a=d=\cos\alpha$, $c=\sin\alpha$, $b=-\sin\alpha$, so $z=\frac{2\cos\alpha+2i\sin\alpha}2=e^{i\alpha}$ and $w=0$. Hence
$$SO(2)\longleftrightarrow\{(e^{i\alpha},0):\alpha\in\mathbb{R}\}.$$

Step 2: linear bijections preserve convex hulls. Let $T:M_2(\mathbb{R})\to\mathbb{C}^2$ be $A\mapsto(z,w)$. $T$ is linear and bijective, so it maps convex combinations to convex combinations, and $T(\operatorname{conv}X)=\operatorname{conv}T(X)$. So we must show $\operatorname{conv}\{(e^{i\alpha},0)\}=\{(z,0):|z|\le1\}$.

Step 3: $\subseteq$. The set $D=\{(z,0):|z|\le1\}$ is convex (the closed unit disc is convex, by the triangle inequality $|\lambda z_1+(1-\lambda)z_2|\le\lambda|z_1|+(1-\lambda)|z_2|\le1$) and contains the circle. So $\operatorname{conv}\{(e^{i\alpha},0)\}\subseteq D$.

Step 4: $\supseteq$. Let $|z|\le1$ and write $z=re^{i\varphi}$ with $r=|z|\in[0,1]$. Then
$$z=\frac{1+r}2\,e^{i\varphi}+\frac{1-r}2\,\bigl(-e^{i\varphi}\bigr).$$
Indeed $\frac{1+r}2-\frac{1-r}2=r$. The two weights are $\ge0$ and sum to $1$, and $e^{i\varphi}$, $-e^{i\varphi}=e^{i(\varphi+\pi)}$ lie on the unit circle. So $(z,0)$ is a convex combination of two points of the circle.

Back in matrices. In real terms: $\operatorname{conv}SO(2)=\left\{\begin{pmatrix}p&-q\\q&p\end{pmatrix}:p^2+q^2\le1\right\}$, and the matrix with $(p,q)=r(\cos\varphi,\sin\varphi)$ equals $\frac{1+r}2R_\varphi+\frac{1-r}2R_{\varphi+\pi}$.

Remark. This hull is only $2$-dimensional. In contrast, $\operatorname{conv}O(2)$ is the whole operator-norm ball $\{\|A\|\le1\}=\{|z|+|w|\le1\}$, which is $4$-dimensional: the improper rotations $\{z=0,|w|=1\}$ are needed to fill it. In particular the $2\times2$ matrices of operator norm $\le1$ cannot be written as convex combinations of rotations alone.

Lesson. Convex hulls are invariant under linear isomorphisms, so choose coordinates in which the set is a familiar one (here a circle).