  Let $A_\N$ be the symmetric operator given by the restriction of $A$ to $\N$,
where $A$ is a self-adjoint operator on the Hilbert space $\H$ and $\N$ is a
linear dense set which is closed with respect to the graph norm on $D(A)$, the
operator domain of $A$. We show that any self-adjoint extension $A_\Theta$ of
$A_\N$ such that $D(A_\Theta)\cap D(A)=\N$ can be additively decomposed by the
sum $A_\Theta=\A+T_\Theta$, where both the operators $\A$ and $T_\Theta$ take
values in the strong dual of $D(A)$. The operator $\A$ is the closed extension
of $A$ to the whole $\H$ whereas $T_\Theta$ is explicitly written in terms of a
(abstract) boundary condition depending on $\N$ and on the extension parameter
$\Theta$, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as
a set) to the deficiency spaces of $A_\N$. The explicit connection with both
Kre\u\i n's resolvent formula and von Neumann's theory of self-adjoint
extensions is given.
