Can $S^n$ be defined without reference to an $n+1$ dimensional embedding space? It seems like the most common definition is the set of all vectors in $\mathbb{R}^{n+1}$ which are unit length. Is this a necessary definition?
If you just want something homeomorphic to $S^n$, you can use the one-point compactification of an open ball in $\Bbb R^n$.
Here's an equivalent definition: $S^n$ is the unique compact, simply connected $n$-dimensional manifold of constant, positive sectional curvature $+1$. That gives you the sphere with the metric you'd get from the embedding in $\mathbb{R}^{n+1}$.