Why is there no explicit formula for the factorial? I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by:

$$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$

However, I know that no such formula is known for the most simple product (in my opinion) - the factorial:

$$n!=\prod_{k=1}^n k = \prod_{k=2}^n k$$

I don't know if that is how products work, but I would really like to know!

So my question is why is there no explicit formula (in terms of n, other than n(n-1)...2*1) for the product of the first n integers? Is there a proof that says that one cannot exist or is it that one has not been discovered?

By explicit formula I mean a non-functional equation that does not require n amount of steps to calculate - just like the summation formula does not require n additions.
(The following is a joke.)

Put $a_n:={1\over 6}n(n+1)(n+2)$ and $b_n:={1\over2}n(n+1)$, and define the magic constant 
$$\xi:=\sum_{n=1}^\infty {n!\over 2^{a_n}}\ =\ 0.630882266676063396815526621896\ldots\quad .$$
Then 
$$n!=\bigl\lfloor\&gt; 2^{a_n}\xi\&gt;\bigr\rfloor\quad \bigl({\rm mod}\ \ 2^{b_n}\bigr)\qquad(n\geq1)\ .$$
Try it out!
If I am not mistaken, Manjul Bhargava generalized the factorial function to include numbers such as (pi)! and (2+√3)!