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 endomorphisms $\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.

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.

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 endomorphisms $\varphi $, one defines the unitary element $$ u_\varphi = \sum_{i=1}\varphi (S_i)S_i^*. $$

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

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 endomorphisms $\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.