Given $S_n$ acting over $T=\{1,2,\dots,n\}$ in the usual way, can I count the number of elements in $S_n-\cup_{i=1}^n G_i$ (where $G_i$ is the stabilizer of the element $i\in T$) and get an explicit form for them (I mean obtain certain elements that generates this set)?
For example: for $n=3$, the elements of $S_3$ are $(1,2), (1,3), (2,3), (1,2,3), (1,3,2), ()$, so the only elements that are in this set are $(1,2,3)$ and $(1,3,2)$.
Yes, these are known as derangements. Using the principle of inclusion-exclusion, you can show that the number of them is $$n!\sum_{k =0}^n \frac{(-1)^k}{k!}\,.$$