If the universe is infinite in space, wouldn't the Big Bang originate from a 2 dimensional plane instead of a single point?

The universe may be infinite or not, we do not know.

No one has yet proven the universe to be finite.

No one has yet proven the universe to be infinite.

Lots of scientists who have pet theories about what MAY have come before the Planck Time, need the universe to be spatially finite, otherwise their pet idea does not work. However, that is not proof that the universe is finite.

The only tests done so far tend to prove:

-- the whole universe is bigger than the portion we can see (at least three times bigger -- which could include infinite)

-- the Observable Universe seems to have a "flat" geometry, which still allows the universe to be infinite in spatial extent. Although there are some finite geometries that could allow the universe to be flat AND finite, the easiest configuration to explain the geometrical flatness is still an infinite universe.

Some confusion comes from the idea that we know "spacetime" (a mathematical description of the HISTORY of the universe) is finite: it has a bound in the past (13.8 billion years ago, when seen from our frame of reference).

"The Big Bang" is not an object nor a moment in time. Big Bang is just an awful nickname for the theory - and it was given by an adversary of the theory (Fred Hoyle) who wanted to remind others that the theory (1948) comes from an old idea (by a priest, 1927) that the universe began in the "explosion" of an atom.

The real theory has no such explosion, nor does it claim the universe started from a point.

The "singularity" that some people see in the Big Bang model, is a moment in time that would come "before" the Planck Time, where some output parameters grow without bounds. Notably, the energy density appears to grow without bound as one goes back in time "before" the Planck Time.

Problem: any time "before" the Planck Time is outside the domain of the model (in other words, the theory cannot be applied to any time before the Planck Time).

For all we know, IF (a big if) there really was a singularity moment, then the unbounded output would belong to the entire universe, regardless of what it looked like at the time:

-- a single point (0 dimensions)

-- a line (1 dimension)

-- a plane (2 dimensions)

-- a volume (3 dimensions)

And any one of these (except the point) could be wrapped around higher dimensions, making it finite but without bound (for example, the line wrapped around as a finite circle within a 3-dimension "space" of some sort).

IF (a big if) the universe is infinite now, then the simplest model (mathematically) is the one where the singularity occupied the entire R^3 volume (itself spatially infinite). However, such a model is the dullest one -- you can't have exciting things happen, like colliding membranes, themselves subspaces of a multi-dimension universe (the multiverse).

The two-dimensional "singularity" you describe is already used in some of the more exciting (however, less likely) models.

The zero-dimension is extensively used by... non-scientists. For example, the Bogdanov brothers had a TV show where they based their entire prelude to the Big Bang as a single point of unbounded density, temperature, etc., to which a single number value was assigned... and the whole universe is the expansion of this point into as many dimensions as you want. Their "scientific sounding" book (God and Science, 1991) was a bestseller at the time. They also published papers for scientific papers on the same topic.

The Bogdanov papers has since been declared of "nil" value, but the idea is still very popular.

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All we can understand (and test) so far, is whatever happens after the Planck Time. At the Planck Time:

-- the initial energy already existed (the theory is silent about its source)

-- space was already expanding in three dimensions (the theory does not even try to explain why - it simply accepts the observation that it is)

-- the energy density was not infinite (there is an actual, finite value)

-- matter did not yet exist (the theory does explain how some of the energy becomes matter).

If the universe is infinite now, then it MUST have been already infinite at the Planck Time.

Before? we have no way of knowing, since we do not understand (yet) how things work at higher energy density levels.

There never was a Big Bang at all. This documentary may enlighten you:

https://www.youtube.com/watch?v=IFFl9S39CTM