Hii, I wanted to make this post since I’ve been having a lot of trouble with all the 大 related vocabulary, specifically 大きい, 大した, 大きさ and 大いに. I always mistake the meaning of one for another and I’m unsure of which one is which.
Would appreciate if anyone could give me some advice or a way of getting them right, thanks 
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大きい is big, and 大きさ is size.
This happens with some other qualitative adjectives:
An easy example would be 長い (long) and 長さ (length) or 高い (high) and 高さ (height).
This could help you relate those two.
The さ means -ness. So 大きさ means “bigness” (size).
From Bunpro it “Converts an Adjective into a Noun”. There’s also み which functions the same, but the two are used in different cases. This site explains the difference better than I could.
http://maggiesensei.com/2017/04/17/the-difference-between-さ-み-sa-mi-suffixes-to-make-a-noun/
I’ve always kept 大した separated by thinking of it as someone who is a big $h!t (ie. someone who is important, or at least thinks they are).
It’s not the main meaning but it it is a synonym and should at least help with reviews.
You also usually hear stuff like 大したことない “It’s not a big deal.” (and a lot more vocabulary) in anime or series, so watching stuff in Japanese will help you with telling stuff apart.
Yeah I watch a lot of anime but I never paid attention to what they actually were saying, now, when that I’ve learned some words I sometimes hear them, though some slip away too hehe. Will try to pay more attention! 
大した - I always think “I did great”, since した makes it “past tense”.
大きさ - さ sounds a little bit like “size”, which is also the meaning.
I hope that can be of some help at least.
-に is a common adverb suffix, so you can often think of it as meaning “with~”: For 大いに I think “with bigness”, i.e. “very”. Not to be confused with 大きく “in a big way” 
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To make it less confusing, you could try thinking of 大いに as 多い + に, and use the “a lot of” definition, which is pretty similar to 多い.
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