Although not exactly what the OP has in mind,
there is another interesting characterization of endomorphisms of $\mathcal O_n$ in terms of  single elements.  Namely
there is a one-to-one correspondence between endomorphisms of $\mathcal O_n$ and unitary elements of $\mathcal O_n$ given
as follows:

 * If $u$ is a unitary element, one defines an endomorphisms $\varphi _u$ by sending each generator $S_i$ to $uS_i$.

 * Conversely, given an endomorphism $\varphi $, one defines the unitary element
  $$
  u_\varphi  = \sum_{i=1}^n\varphi (S_i)S_i^*.
  $$

It is in fact very easy to show that these correspondences are each other's inverse.