Let $(\mu_{\alpha})_{\alpha\in\mathbb{N}_0^d}$ be a collection of signed/complex Radon measures on $U\subset \mathbb{R}^d$. Let's call it locally finite if for every compact $K\subset U$, $|\mu_{\alpha}|(K)=0$ except for finitely many multiindices $\alpha$'s. Now given such a collection and for a test function in $f\in \mathcal{D}(U)$ define $$ \phi(f)=\sum_{\alpha\in\mathbb{N}_0^d} \int_U \partial^{\alpha}f(x) \ d\mu_{\alpha}(x)\ . $$ This $\phi$ is in $\mathcal{D}'(U)$ and I am pretty sure every element on $\mathcal{D}$ looks like this. The writing is of course very non unique because of the redundancy between derivatives of the same function. This is just an elaboration on Isset's answer. I think it is nice to think of all these "layers" index by $\alpha$ simultaneously. Take the embedding $U\rightarrow \mathcal{D}(U)$ which sends a point to the corresponding Dirac unit mass. It is in the "top layer" $\alpha=0$. What about making this embedding smooth? By taking Gateau derivatives, you see that you start cascading down to deeper and deeper alphas. This is also in accordance with the physical intuition behind dipoles,...multipolar charge distributions which were a source of inspiration for L. Schwartz. BTW Hahn-Banach is not needed. One can use truncation/convolution to reduce to $\mathcal{S}'$ and the Hermite function basis (see Barry Simon's JMP article).