Why is $SO(n)$ a smooth manifold

Question

I am new to differential topology and a question that I came across was to show that $SO(n)$ is a smooth manifold of dimension $\frac{n(n-1)}{2}$. The dimension follows from the fact that the columns are orthonormal, hence the first column belongs to $S^{n-1}$ and the second to $S^{n-2}$ etc. I don't quite have much intuition yet but I'm finding it very difficult to construct an atlas and show that the transition functions are smooth. I don't really understand what the open sets of $SO(n)$ would be or where to map them in $\mathbb{R}^\frac{n(n-1)}{2}$. Any tips or suggestions for other ways to show smoothness? For reference I haven't seen too much other than what smooth manifolds are and a few properties about tangent vectors and differentials (which I don't understand much of yet).