A famous factorization is $(1+X_1) (1+X_2) \cdots (1+X_n)$ where $X_1=0$, $X_k=
(1,k)+(2,k)+\cdots +(k-1,k)$ for $2\leq k\leq n$. $X_k$ is called a
*Jucys-Murphy element*, though this factorization is due to Alfred
Young in 1902. Jucys gave the $q$-analogue
  $$ (q+X_1)(q+X_2)\cdots(q+X_n) =\sum_{\pi\in S_n} q^{c(\pi)}\pi,
  $$
where $c(\pi)$ denotes the number of cycles of $\pi$. See for instance
https://en.wikipedia.org/wiki/Jucys-Murphy\_element.

Another useful factorization is $T_2T_3\cdots T_n$, where
  $$ T_k = \sum_{j=1}^k (k,k-1,\dots,k-j+1). $$
This also has a $q$-analogue:
 $$ T_2(q)T_3(q)\cdots T_n(q) = \sum_{\pi\in S_n}
  q^{\mathrm{inv}(\pi)} \pi, $$
where $\mathrm{inv}(\pi)$ is the number of inversions of $\pi$ and
  $$ T_k(q) = \sum_{j=1}^k q^{j-1}(k,k-1,\dots,k-j+1). $$
Zagier in 1992 found a deep factorization of $T_k(q)$ which was used
by Philip Hanlon and me in
http://math.mit.edu/~rstan/papers/qdef.pdf (see Theorem 2.1).