In this paper, we determine the structure of $KO$-cohomology of complex projective space $\bc P^l$ and its product space $\bc P^l\times\bc P^m$ as algebras over the coefficient ring $KO^*$. We also give a description of the map $KO^*(\bc P^{l+m})\to KO^*(\bc P^l\times\bc P^m)$ induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product $KO^*(\bc P^l)\otimes_{KO^*}KO^*(\bc P^m)\to KO^*(\bc P^l\times\bc P^m)$ to see that non-existence of the formal group structure on $KO^*(\bc P^{\infty})$.