When looking for synonyms for the phrase "big cage," there are a variety of options at your disposal. Some alternatives to signify a large enclosure for animals include "spacious aviary," "expansive coop," or "roomy kennel." For those seeking to describe a bigger, more open-air enclosure, you might opt for "large pen," "vast enclosure," or "grand outer area." Additionally, for specific types of animals, there are alternative phrases such as "giant terrarium" for reptiles or "massive aquarium" for fish. Ultimately, the synonym you choose to describe a large, enclosed area for animals should depend on the specific context and intended connotation of the given situation.