A property of orthogonal matrix

Question

suppose two orthogonal matrix $O_1, O_2 \in \mathbb{R}^{2 \times 2}$ in SO(2), consider $x = \text{Tr}(O_1O_2)$, we know that $$O_1 = \begin{pmatrix}\cos(\theta_1) &\sin(\theta_1) \\ -\sin(\theta_1) & \cos(\theta_1)\end{pmatrix}, \quad O_2 = \begin{pmatrix}\cos(\theta_2) &\sin(\theta_2) \\ -\sin(\theta_2) & \cos(\theta_2)\end{pmatrix}$$ for some anlges $\theta_1$ and $\theta_2$. Then we have $$ x = 2(\cos(\theta_1)\cos(\theta_2) + \sin(\theta_1)\sin(\theta_2)) = 2\cos(\theta_1 - \theta_2).$$ which is just the $2\cos(\theta)$ that $\theta = \theta_1 - \theta_2$. My question is that, does similar proproty exists for higher dimension $n$? Let's say $O_1, O_2 \in \text{SO}(n)$, does any property or simple expression of $\text{Tr}(O_1O_2)$ exist?