Would it be correct to say that ozone is an element?
It would be more correct to describe ozone as an allotrope of oxygen. Dioxygen (about 1 in five of the molecules making up air) is another allotrope.

The periodic table describes the atoms that make up elements as this is the most fundamental property. Many other elements, though, exist in multiple allotropes. Carbon, for example, has several: graphite, diamond, buckminsterfullerene and quite a few others. 

All allotropes consist of only one type of atom but the bulk physical properties depend on how the atoms are joined up (flat plates of carbon hexagons with multiple bonding in graphite, a three-dimensional network of singly-bonded tetrahedral carbons in diamond). 

We don't normally add too much about allotropes in a periodic table as it would make the table a little big, although some include limited information about how many there are. And, usually, only one is thermodynamically stable under normal conditions (so the physical properties will be about that one).