Forumula for the product of the first 'n' numbers?

I know that we are able to derive a formula for the sum of the first 'n' positive natural numbers, that sum being:

I know that we can derive this formula by writing out the sum as

and doing some algebraic manipulation. But, what if we have the following:

Is there a way in which we can manipulate the product expansion (simmilar to the way in which we can manipulate the summation expansion) which will cause us to arrive at a direct formula for the product of the first 'n' positive integers simmilar to

such that we'd have something like the following

[p.s. also, I realize that would denote the function I am refering to, but that sort if "notation" is not what I am refering to here. I'm asking weather there exists an explicit forumula for the product which does not have to be written in the form or any form that includes the ".....", but rather a **precise** **formula **simmilar to the one used for the sum of the first "n" positive integers.]