Is there any known formula for multiplying out factorials?
$x$ being a positive integer, $x!$ is defined as $$x(x-1)(x-2)\cdots3\cdot2\cdot1$$
My question is if there exists another "multiplied out" form of that product, as a sum.
I tried multiplying out the first three, four, five factors and seemingly random coefficients that seemed to get bigger and bigger the more factors I multiplied out appeared.
Yes, indeed. This has to do with something called Stirling numbers of the first kind. The numbers you are looking at are $$n!=n(n-1)\cdots (n-n+1)=\sum _{k=0}^n(-1)^{n-k}{n\brack k}n^k,$$ where ${n\brack k}$ is the number of permutations of $n$ having $k$ cycles.