The word "aromatic" is most commonly used for systems that do have rings, but an enhanced resonance stabilization with delocalized $\pi$ bonding can occur without rings whether we apply the label "aromatic" or not.
Carbon dioxide is actually the first of a series of linear molecules having the formula $\ce{C_nO2}$ or $\ce{O=C=C=...=C=O}$, in which all the pairs of neighboring atoms are joined by double bonds and thus we have a linear molecule.
If we draw out the molecular orbitals of this structure we discover that the molecule will have a closed-shell structure when there are $4n$ $\pi$ electrons (not $4n+2$), and carbon dioxide with eight ($n=2$) has all the hallmarks of a system with extra $\pi$ electron stabilization and delocalization (see here for a discussion). Similarly, the $12\pi$ member carbon suboxide, $\ce{C3O2}$, formally has cumulated double bonds: but its carbon-carbon and carbon-oxygen bond lengths are shorter than those in ethylene and formaldehyde respectively. Carbon suboxide is prone to polymerization but can be kept as molecular $\ce{C3O2}$ under carefully controlled, dark conditions. By contrast, ethylene dione, $\ce{C2O2}$, with an unfavorable electron count for forming a closed shell, decomposes on a nanosecond time scale.