I've drawn a structure of $\ce{N2O2}$ like this:
$$\ce{:\overset{\large.\!\!.}{\underset{\large.\!\!.}{O}}-N#N-\overset{\large.\!\!.}{\underset{\large.\!\!.}{O}}:}$$
I don't know if it's possible. Please correct me if I'm wrong. Also, what would be the nature of it?
If you're just looking at the Lewis Structure from the perspective of the octet rule, it does appear that the structure is correct. Dinitrogen always has a lone pair of electrons which could conceivably be used for dative bonding as you suggest. So from that perspective there appears to be nothing wrong at all - other than that it doesn't exist in nature in this way.
If you're interested in learning about the dimer, a number of proposals have been made in this 1994 JACS paper. If you're interested in the hyponitrite anion, you can read about it on Wikipedia (always a good place to start), and for your reference the structure of the Sodium salt is below:
It's wrong in the sense that it is inconsistent with what is found experimentally, according to Structure of the NO Dimer J. Am. Chem. Soc., 1982, 104 (17), pp 4715–4716.
The experimental structure is cis-ONNO, with ONN angle of 99.6 degrees.
The N-N bond length is 2.236 Angstroms, while the N-O bond lengths are 1.161 Angstroms.
No single Lewis structure can accurately describe the electronic state of this molecule; however, considering the bond lengths, a single N-N bond and triple N-O bonds would be better.
See The 16 valence electronic states of nitric oxide dimer form more information.
