In depictions, recreations and in general displays of the Roman armour, it usually doesn't cover the arms very well. I'm curious as to why this was chosen? Obviously, covering it with plate would increase the cost of armour quite a bit to each soldier. But what about covering it up with fabric, at least that would be helpful against cold and as a very, very light protection against common hazards that clothing prevents. In medieval times, there wasn't much sleevelss armour, it seems. Am I completely wrong?
TL;DR: Why did the Romans', and so much other armour, go sleeveless?
The Romans did occasionally wear arm protection when it suited their needs on campaign. There are depictions of legionnaires wearing arm protection in Dacia (known as the Manica) but it is not known how common it actually would have been among the soldiery. It was particularly useful for protection against scythes or other common farming tools used as weapons.
As for wearing fabric, this would actually have been quite detrimental. Arms would typically only sustain minor cuts (deep, but not lethal) and disease would be the worrying factor with wounds that were anywhere but the head and torso. By putting fabric over the arm, when a cut is inflicted the fabric, which is exceptionally dirty on campaign, would get into the wound and fester and infect the wound. Nonetheless, in the Northern areas along the Rhine soldiers did wear fabric on their arms and even pants, which were seen as most barbaric but eventually the soldiery use of pants spread throughout the Empire.
TL;DR: They did occasionally wear armor on arms, clothing on arms causes diseases.