The set of traces of orthogonal matrices is compact

Besides using compactness of the set of orthogonal matrices, you can show directly that the set of possible traces is $[-n,n]$. Note that $I$ and $-I$ are orthogonal with $\text{Tr}(I) = n$ and $\text{Tr}(-I) = -n$. On the other hand, each matrix element of an orthogonal matrix has absolute value at most $1$, so $\text{Tr}(T) = \sum_{i=1}^n T_{ii}$ has absolute value at most $n$.

To get orthogonal matrices with every trace value from $-n$ to $n$, consider those made from diagonal blocks of the form $$\pmatrix{\cos \theta & \sin \theta\cr -\sin \theta & \cos\theta}$$ with an additional diagonal entry of $+1$ or $-1$ in case $n$ is odd.