An element is the smallest entity that has it's own characteristics in Chemistry. Any two different elements differ fundamentally in respect to the number of protons in their nuclei. An element is always of single atom. It's the characteristic properties of that element that would decide in what form or kind of molecule it will be found in nature.
Diatomic element is a kind of molecule in which two atoms of the same or different elements have combined chemically to form a molecule. If we restrict ourselves to the context of elements, then diatomic element is one among various allotropes of the element.
Oxygen is found in nature as diatomic and triatomic (ozone). The reason we don't find monoatomic oxygen often around is that an atom of oxygen is highly reactive. You may have heard of nascent oxygen but it exist for very short time in some chemical reactions. O, O2 and O3 are different allotropes of the same element oxygen. Besides these, there are some other allotropes of oxygen that are listed in the given reference.