In Thor: The Dark World, it's implied that long ago, the Dark Elves turned an Infinity Stone into the Aether. In Guardians of the Galaxy, we see an
Infinity Stone in its natural state, as described by Odin.
The Tesseract seems to not be a plain Infinity Stone, it was crafted into the familiar cube we've seen in several Marvel films.
Who turned the Space Stone into the Tesseract?
In the Marvel Cinematic Universe, the Infinity Stones appear to have very different physical appearances unlike their comic versions which all appear in the form of small gemstones.
Just as there is no particular reason for the gems to appear as gemstones given their origins as Nemesis (the remnants of a dying godlike being who supposedly died from loneliness) their appearance was not guaranteed.
Each gem is a small, smooth oval, and known by their respective powers. The gems are indestructible but not immutable. On two occasions one or more of the gems have appeared in the form of deep pink spheres that were several feet in diameter.
On other occasions the gems have appeared in their normal shape but with different coloring (i.e. when the Soul Gem appeared to be red when worn by the Gardener).
In the Ultraverse, the merged gems, Nemesis, were once again separated after a climactic battle with Ultraforce and the Avengers. The space, power, soul, and mind gems morph into humanoid-like beings. The accounts of these beings are not known, as well as how these beings returned to their "gem" forms.
Nemesis in her restored form
Given the divergent appearances of the Infinity Stones in the Marvel Cinematic Universe, it may simply be a need for visual differentiation, making each stone unique in their appearance for the viewing audience.
Considering their near-all powerful nature and possible sentience, their shape and overall appearance should not be difficult to change as desired.
*Loki's staff is still in contention as the Mind Gem.
