The skeleton description says that while it cannot speak it, it can understand the languages it knew in life. Can it not speak because of its undeath or could it learn to speak a language?
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By RAW default, no.
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As explained by others it has no basic, default, means of expressing language. It's not about learning the language, it's the capability of the spell/energy powering the skeleton(s). Undeath is not in itself an impediment to speech (see: Liches).

Therefore you need to extrapolate a bit. The RAW description quoted above seems to imply the power source provides bare-minimum world interaction capabilities. And does not extend to advanced capabilities like mimicking a voice (disembodied or otherwise).

But, the "talking skull/skeleton" trope is incredibly common. So...

By customization, yes. (Possibly with the aid of a magic item.)
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It is your game, after all. And creativity is a core value of the gaming experience.

A skeleton is already magically animated, so if you wanted a **specific skeleton** to have the ability to speak, there are several creative ways you could explain the phenomenon.

* A magic item equipped by its creator, or even left over from its previous life. (An enchanted ring or amulet?)
* A slightly more powerful skeleton who gained the ability to speak over time via wild magic or the like.
* A skeleton created with extra riders on the creation spell by its creator. So maybe it was imbued with a secondary spell to allow a disembodied voice.

You could have all skeletons speak of that's how you want to run your world, but that'd be your own custom flavoring.

A speaking skeleton, otherwise, would be **special for some reason**.
It is true, on pg 272 of the 5e Monster Manual, it states:

> 
> They can't read, speak, emote, or communicate in any way except to nod, shake their heads, or point.
> 
> 
> 

... and D&D Beyond - [Skeleton entry](https://www.dndbeyond.com/monsters/skeleton) states ...

> 
> Understands all languages it knew in life but can't speak
> 
> 
> 

... but both of these interpretations seem to be relatively 'new' 5e expansions to d&d skeleton lore. Wrt Habitual Behaviors in the 5e MM, if "skeletons of nobles might continue in an eternally unfinished dance" && "can load and fire a catapult or trebuchet".
Also, wrt D&D Beyond, it appears to be poorly worded. If taken literally, that it simply cannot speak; then it would be able to use sign language, portions of thieves cant, or write & still communicate.

I think all of the older versions are more consistent wrt skeletons in this respect; in that the creator can command them via magic (and it ignores the medium of the command.)

Therefore we need to break up your question:

1. Can it not speak because of its undeath?

If you look at all the lore in total, the answer here would be the skeleton cannot "communicate", speaking or otherwise. If you look at 5e literally, then the two sources are possibly contradictory & you'll have to subjectively decide for your world/campaign.

2. Could it learn to speak a language?

The simple answer is no... however, again looking at all the lore, should a different necromancer, demon, etc gain control of the skeleton; the method of control would give the ability to command irrespective of language.

Relatedly, an item of consideration is whether or not skeletons are deaf, & how their "vison" works.?. In my campaigns, skeletons are "unintelligent", and are deaf; with their "vision" being more animalistic/primitive in nature (they can't read, or understand complex items <missle weapons & doors included>)...

This isn't to say a specific necromancer couldn't craft a "unique breed" of skeletons that have different features, including higher intelligence; but just that the "average" skeleton does not.