This book may (or may not!) be overkill, but I've been reading Matrix Groups for Undergraduate (2nd Edition) by Tapp. It's been going over all the matrix groups including SU. I don't remember if it went into detail on U(3) or not, but it talks in depth about the matrix groups! I enjoy the writing style. Sorry, it's been a while since I was working on it so it's hard for me to remember too many specifics!
A math book might do U(n) and then tell you to specialize n to 3 just like the common joke of how to visualize 4 dimensions and the mathematician doing the overkill R^n first and then saying then let n=4.

So in this case where it looks like you want explicit formulas and numbers, you can go with an intro to group theory for particle physics. It will likely have a special section just for SU(3). There is one by Georgi. I don't remember that book's chapters, but it is likely there in the explicit and specialized to n=3 form that you seem to want.
Yvette Kosmann-Schwarzbach: *Groups and Symmetries — From Finite Groups to Lie Groups*

Is doing SU(3) & quarks at the end

ED: A lot easier to understand:

Andrew Baker: *Matrix Groups — An Introduction to Lie Group Theory*
Hi,

I just wrote a hugely long and detailed explanation of something about this but i decided to delete it and just give the TL;DR. That SU\_3 is hard to understand either separately as a complex or real object, but both SU\_3and SL\_3(R) are subgroups of SL\_3(C), and what people some do is intentionally use the 'wrong' subgroup and hope that not much has gone wrong. So, although it is weird, pretend you are using SL\_3(R) instead of SU\_3, and get your answer and then check if by luck it is still correct. SL\_3(R) and SU\_3(C) are two "forms" of the same group. you can look up "forms" and "descent", and that is a more general and sort-of beautiful theory too.

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Actually google isn't useful, but JP Serre described "forms" and "descent" beautifully somewhere, maybe in 'groupes algebriques et corps de classes'

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I can't resist giving one more hint, that SL\_3 acts on the 'flag' variety which is configurations of a line in a plane in C\^3, which is the subset of P\^2 x P\^2 defined as \{ ([a:b:c],[d:e:f]: ad+be+cf=0}. So any subgroup of SL\_3 acts on any vector-space naturally associated to that variety. The equivariant line bundles on that variety are parametrized by particular pairs (i,j) of integers and their vector spaces of global sections are natural vector spaces that all these groups act on. This is how all complex irreducible representatinos of the real Lie algebra SU\_3 arise I think, and this sort-of supports the notion that it is OK to use the 'wrong' form.

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To get the real representations I think you just delete the subset that aren't complexifications of a real representation.

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I'm not sayig you were asking about using any but the usual 3x3 matrices, but I guess the point is that the actual matrices in the 'ordinary' matrix represenation are mysterious or "they are what they are and what else can we even say about them",  so people try changing bases and stuff and trying to find easy and convenient matrix representations.

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