Informally asking, can we step through all permutations of the set $\{1,\ldots,n\}$ by just using transpositions?

More formally: For any $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$ and let $S_n$ be the set of all bijections (permutations) $\pi:[n]\to [n]$. For any set $X$ let $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big\}$. We let $\pi,\psi\in S_n$ be connected by an edge if "they are one transposition away from each other", or more formally, set $$E_n = \big\{\{\pi,\psi\}\in [S_n]^2:\exists a<b\in[n]:\psi = (a\;\;b)\circ\pi\big\}.$$

For what $n\in\mathbb{N}$ does the graph $(S_n, E_n)$ allow for a Hamiltonian cycle? Or at least for a Hamiltonian path?