It took mankind tens of thousands of years to figure out that air existed, let alone that it was matter. It was only in recent human history that we figured out anything about air. Proving that air is matter is analogous to today's physics experiments where you cannot see the object of your study, but have to define its properties and its existence from indirect evidence.
We define matter as something which occupies space, is effected by gravity and has weight. Make a vessel that won't collapse if there is no air inside of it. Weigh the vessel when it is full of air. Then pump all of the air out and weigh the vessel again. The difference in weight is the weight of the air.
There is a famous experiment done by Otto Von Guericke in 1654 in Regensburg, Germany. Regensburg was a Roman outpost on the banks of the Danube River. If you ever go there, I highly recommend the Wirstkuke, an 850 year old restaurant near the river. It was there when Guericke was studying air and he may have had a dinner or two there. Anyway, to prove that air exists and has pressure he made a hollow sphere made of two copper halves and sealed it with a gasket. He used an air pump, which he also invented, to pump the air out of the sphere. Air pressure held the two halves of the sphere together. He then took two teams of horses and had them try to pull the sphere apart. They failed. Guericke then opened a valve that let the air back in and that is when the sphere fell apart under its own weight. The sphere was 14" in diameter meaning the air pressure exerted a force of approximately 4.5 tons.
The force would have been the same if one side of the sphere was attached to something fixed, like a really big rock, instead of another team of horses. Guericke might not have understood that or he might have just appreciated the drama of using two teams. Showmanship, you see, is important even in science.