For unions we have the generalised axiom, not just union for pairs.
But for pairing we don't have generalisation, that a collection exists for any number of sets.
If we have some axiom that says given any number of sets we have a set that contains all of them as its members, what will go wrong here?
On
It depends on what you mean by number of sets. If it is a finite natural number, then you can get it by the axioms of pairing and induction. If you want possibly infinitely many sets, then you either already have some set containing all of those sets as elements, or you have some sequence of sets, namely a function on an index set, in which case what you want follows from the axiom of replacement. If you have neither, then just where do you get your sets from?
On
You can get any finite set of sets from the pairing axiom and unions.
In the language of set theory, it is difficult to write the general axiom formally, but we can prove individual cases as needed just from pairing and unions.
Basically, there is no need for it, and it adds a complicated "axiom scheme" to our theory.
http://en.m.wikipedia.org/wiki/Axiom_of_pairing#Generalisation
How would you write it?
An axiom is not just a philosophical assertion - it is a precise, formalizable statement. "Any number of sets can be combined into one" isn't really precise.
The problem is the word "number." What do you mean, "any number of sets?" One way to approach this is to say, "Whenever I have a set $X$, and $X$ 'counts' some other family of sets (that is, I can associate to each element of $X$ a new set $A$), then the set of those sets exists." This can be made precise, but isn't new: it's the axiom of Collection!
Another way to approach the problem is to consider allowing "axioms" which are infinitely long - but then we leave the realm of first-order logic well behind, and set theory starts creeping into our axiomatization of set theory in odd ways.