"There is a big library near my house."
Ookii is being used. It is what is referred to as a na-adjective. When it is placed before the noun it is describing it becomes おおきな.
「おおきなとしょかん」'A big library.'
「としょかんはおおきいです。」'The library is big.'
Here is a list of some na-adjectives for beginner Japanese.
There is some inaccuracy here. 大きい and 大きな are 2 separate words although both of them means big. 大きい is not a Na-adjective and it changes form according to I-adjective rules.
The difference in usage is 大きな is only used as an attributive adjective before a noun whilst 大きい is used also in other functions, and changes accordingly.
Which one to use before a noun is a matter of Japanese usage habit. Usually 大きな is used on conceptual, intangible matters while 大きい is used on physical things. It seems though a lot of overlapping exist and using either one should not be considered wrong. So yes, 大きい can also be used.
Maybe it'll help when you realize that the sentence could have been written as 私の家の近くには大きな図書館があります。
More than that, though, in simple statements が is pretty much universally used in conjunction with ある.