Embedding torus in Euclidean space For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?

As Aaron Mazel-Gee's comment indicates, this follows from induction. Although you only asked about $n > 2$, it actually holds for $n \geq 1$.
The base case is $n = 1$, i.e. $S^1$ embeds in $\mathbb{R}^2$, which is clear
For the inductive step, suppose that $T^{k-1}$ embeds in $\mathbb{R}^k$. Then $T^k = T^{k-1}\times S^1$ embeds in $\mathbb{R}^k\times S^1$ which is diffeomorphic to $\mathbb{R}^{k-1}\times(\mathbb{R}\times S^1) \cong \mathbb{R}^{k-1}\times(\mathbb{R}^2\setminus\{(0, 0)\})$. Now note that $\mathbb{R}^{k-1}\times(\mathbb{R}^2\setminus\{(0, 0)\})$ embeds into $\mathbb{R}^{k-1}\times\mathbb{R}^2 \cong \mathbb{R}^{k+1}$. As the composition of embeddings is again an embedding, we see that $T^k$ embeds in $\mathbb{R}^{k+1}$.
By induction, $T^n$ embeds in $\mathbb{R}^{n+1}$ for every $n \geq 1$.
