There is no precise, clear cut answer to your question, I'm afraid, though I can offer some information to perhaps put it into context. Whether something is a skirmish or a battle is largely a matter of perspective, and a threefold one when you consider that a single battle can last long enough and cover a large enough geographical area for an unbroken string of combat to graduate to being christened to a Campaign in its own right.

Generally, in my own experience and knowledge, a vague definition can be gained only by context. If we consider the Battle to be the starting point of decision in human conflict--and given that even large campaigns tend to be defined by battles as decision points for all those engaged, whether victorious or defeated--then a skirmish would logically fall beneath this as a subpoint, but this definition does not fit perfectly either, as many battles are themselves made up of smaller independent actions which in turn define the battle's course. Certainly the Battle of Kursk is considered a single battle, but that battle raged for nearly 2 months and involved more men and material than almost any entire war before the 20th century, much less a campaign or battle. Even a single day in a single location could be a vast battle in its own right: the single day of the Battle of Prekhorovka was a tank battle four times larger than the largest tank battle of the western front(not including North Africa) but is only a small bullet point on the list of decision points for the engagement.

Scaling down, small unit actions can be held apart as their own name-christened battles, with as few as a dozen or less on each side, or hundreds or even thousands if the overall battle is large enough. Likewise, if terrain is constricting or weather is particularly foul, even large engagements can seem to those involved to be nothing more than random hasty gunbattles without rhyme or reason when command and control breaks down.

In the end, trying to decide whether something is a battle or a skirmish is trying to put defined labels on inherently nebulous events, just as trying to decide just exactly where on the color spectrum 'red' ends and 'yellow' begins. You know, generally, where the ends of the spectrum are, and you might even be able to slice a section where you are reasonably certain the answer lies, but beyond that it is open to interpretation.