So is an adverb in that sentence. It's modifying big, an adjective.
Adjective
In the above sentence, "run" is the object of the preposition. It is part of the prepositional phrase "for a run."
"Big" can be used as an adjective.
In the sentence, "balloon" is a noun as it is referring to a person, place, thing, or idea.
In my neighbourhood a big mansion is visible behind all the small houses. Mansion (definition): Villa, big house
It is an adjective, as you can see variety means big.
our school is very big
The man was confident and felt ardor when he went up for his big speech.
Superior can be a noun or an adjective.Examples:He was considered a superior because he had a big house.In the sentence above, superior is used a noun meaning, "a person higher in rank."He had superior knowledge of the inside of the strange house because he had been in it several times.In the sentence above, superior is used as an adjective meaning, "more than average."
a great big gorilla
Biggest is an adjective. It's the superlative form of big.
Big is a predicate adjective. Any word on the other side of the word is or its equivalent when the word is does not happen to be used as a helping verb is either a predicate nominative (noun) or adjective. Jane is a girl. Girl is a noun. Jane is pretty. Pretty is an adjective. (That is one quick way to tell if a word is an adjective!