I had this idea after looking around and seeing that the Tesseract is one of the infinity stones, specifically the space stone. That is how they are able to create the portal in The Avengers. In Captain America: The First Avenger, when Red Skull uses Tesseract-based weapons and fires them at people, shouldn't those people travel through space? With that same logic, shouldn't Red Skull himself travel through space?
The Tesseract is the Space Stone, and as such enables instantaneous transportation to anywhere in space, provided one understands how it works. It is also an excellent power source. Red Skull's experiments with the Tesseract revolved around its way of producing energy, and enabled him to develop energy weapons utilising similar power-generation. That is not, however, the same as the other special abilities the Tesseract possesses, such as instantaneous travel. That is why none of the weapons HYDRA created based on the Tesseract possessed similar characteristics.
As an aside, however, since Red Skull disappears at the end of Captain America: The First Avenger, through a portal created by the Tesseract, he may well reappear at some future date with abilities gained from it, or from someone or something (Thanos, perhaps) at the other end of his journey. Red Skull was originally supposed to be one of two antagonists in the original draft of The Avengers, until a re-write gave Loki the job on his own.