Embedding between Lie Groups
Solution 1:
One needs to come up with a map here, then verify it is an embedding. Fix a basis for $\mathbb{R}^n$. The map $$F: A \mapsto \begin{bmatrix} \det(A) & 0 \\ 0 & A\end{bmatrix}$$ is an injective smooth immersion $O(n) \to SO(n+1)$. As $O(n)$ is compact, all injective smooth immersions $O(n) \to N$, where $N$ is a smooth manifold with or without boundary, are embeddings (this useful fact can be found in chapter 4 of Lee's Introduction to Smooth Manifolds).