If my algorithm has complexity O(n!*n), can I just write O(n!), or do I have to keep it like O(n!*n)?
No. You can't. As $\lim_{n\to\infty} \frac{n!}{n! \times n} = 0 $. Hence, $n! \times n  = \omega(n!)$ or $n! = o(n\times n!)$ (little-oh).