Constructing $E_8$ from its branching to $A_8$
An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here). In particular, in section 4, he describes various possible branchings, and Example 4.4 deals with the branching $A_8 \subset E_8$.
The specific example is rather short, but he includes some further references that might be helpful:
Hans Freudenthal, "Sur le groupe exceptionnel $E_8$", Nederl. Akad. Wetensch. Proc. Ser. A 56 (=Indag. Math. 15) (1953), 95–98 (=pages 284–287 in his Selecta published by the EMS in 2009).
William Fulton & Joe Harris, Representation Theory: A First Course (Springer 1991, GTM 129), exercise 22.21 on page 361.
John Faulkner, "Some forms of exceptional Lie algebras", Comm. Algebra 42 (2014), 4854–4873 (=arXiv:1305.0746).
[Freudenthal's paper is completely explicit and elementary; Fulton & Harris provide a bit more theoretical background; Faulkner is more general and works over an arbitrary commutative ring. —Gro-Tsen]
Of course, the original description of this branching is due to Élie Cartan, himself. For example, see Chapitre IX of his 1914 paper Les groupes réels simples, finis et continu (Annales scientifiques de l'É.N.S. 3rd serie, 31 (1914), 263–355).
His chapter on $\mathrm{E}_8$ starts on page 328, and it is very clear and straightforward. The formulae he gives, both for the complex and real forms, are exactly what you are looking for. I think it's the best reference for this, myself. It's available for free download from Nundam, but I don't have the link in front of me right now.