Haemoglobin is composed of two components : soluble haem(iron part) and insoluble globin(protein part). Even though it is composed of two components, haemoglobin behaves as a single species and takes part in reactions only as a complete molecule, hameoglobin(Hb).
When haemoglobin reacts with oxygen, "oxyhaemoglobin"(Hb.4O₂) is formed.
Oxygen is carried by the RBCs in the form of oxyhaemoglobin.
Haemoglobin never gives away its iron content to oxygen to form iron oxide.
The conditions maintained in the blood aren't favorable for the formation of Iron oxide. Iron oxide is generally formed in the presence of atmospheric oxygen.
Moreover, iron oxide contains iron in Fe⁺³ state and the iron in haemoglobin is normally present in Fe⁺² state.