Here in the US, it's extremely common to shorten the full name to just "House" or "The House". Context makes it clear to anyone familiar with the US system of government. Of course in written usage, capitalization also helps clarify.
Similary, the United States Senate is referred to as "The Senate" or "Senate", although you don't really save as much as it's relatively short already.
If you have a question about the English language and would like to ask one of our many English teachers and language experts, please click the button below to let us know: