can I use the definition of adjoint matrix to find the adjoint of an operator?

Question

i have an operator and i need to find his adjoint, the operator is: $$U_{m,n}=|O_m\rangle\langle O_n|$$ where i can say that $$|O_m\rangle=A_{m,1}$$ and $$\langle O_n|=B_{1,n}$$ so i can say that $$ U_{m,n}=A_{m,1}.B_{1,n} $$ $$ (U_{m,n})^*=(A_{m,1}.B_{1,n})^*$$ I know the properties for the adjunct, and so I can build on that, but I'm not sure if I can do this manipulation with the operator to find its adjunct