The concept of a conformal Killing $p$-form in a Riemannian manifold of dimension $m>p\ge1$ was introduced by S. Tashibana and T. Kashiwada. They generalized some results of a conformal Killing vector field to a conformal Killing $p$-form. In this paper we define a conformal Killing $p$-form with the help of natural differental operators on Riemannian manifolds and representations of orthogonal groups. Then we consider the vector space $\mathbf T^p(M,\mathbf R)$ of conformal Killing $p$-forms and it's two subspaces $\mathbf K^p(M,\mathbf R)$ of coclosed conformal Killing $p$-forms and $\mathbf P^p(M,\mathbf R)$ of closed conformal Killing $p$-forms. In particular, we generalize some local and global results of Tashibana and Kashiwada about a conformal Killing and Killing $p$-forms. In the end of the paper we give an interesting application to Hermitian geometry.