If the question is asking whether there is a definition that encapsulates our universe, then I believe the answer is No. This is because encapsulating a "space" into a formal system requires defining bounds. However, we don't know the bounds of our own universe--let alone what bounds might be possible for any universe.
We can only describe what we can detect in the observable universe. From this we can immediately introduce 2 problems to creating a definition of the universe.
1. We have a limited range of the universe that we have the ability to analyze.
2. We can only analyze things that we have the ability to detect.
As a bizarre example: If the universe is like a balloon and the walls of the balloon were made of a different substance than everything inside of the balloon, would we be able to detect the walls if we don't know what to look for? You might think we'd see an "edge" or a "boundary", but not necessarily, if the universe is say, a torus.
Then we can ask questions about the properties of a physical boundary. Is it permeable? What is on the other side? Does there need to be an "other side"? And these questions don't even require defining whether or not a multiverse is possible.
In mathematics, topology is often brought up when formally describing spaces:
Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior.
The only one of those last 3 that we can measure in our universe is the interior. Ultimately it may be the only relevant one when related to a universe. But then again, we don't have further proof to say "there is no boundary to our universe", lending more credence to no formal definition existing.
Others might argue that we can define "enough" of the universe by putting values to the physical constants we observe. These constants combine to create the shapes and interactions of the observable pieces of our universe.
There are also alternative theories about how the universe might have been created, which would radically change our definition of the universe. From M-theory, for example
Heterotic M-theory has been used to construct models of brane cosmology in which the observable universe is thought to exist on a brane in a higher dimensional ambient space. 781
It has also spawned alternative theories of the early universe that do not rely on the theory of cosmic inflation.
And you could throw yet another wrench into the mix by adding time. What was the universe at time=0? Could it collapse (even if it is found to be speeding up right now)?
Or get really whacky and ask whether or not our universe could split like a biological cell (link) (link). What would the formal definition of our universe look like then? I don't want to go into such ideas, but they do show why adding the "time" of spacetime can drastically change your definition, since we can't observe it from beginning to end (that we know of).
But as an optimist, maybe my answer should be "not yet", or "you can get close for practical purposes". Any field might create a "good enough" version of a definition for solving problems in that field. If creating a formal definition requires omniscience, then I'd move my answer back to a firm "No".
Just for completeness, I feel the need to add a few pieces from Dictionary.com's definitions:
1. the totality of known or supposed objects and phenomena throughout space; the cosmos; macrocosm.
2. Also called universe of discourse. Logic. the aggregate of all the objects, attributes, and relations assumed or implied in a given discussion.
3. Also called universal set. Mathematics. the set of all elements under discussion for a given problem. 781