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)][1].
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](http://www.tandfonline.com/doi/abs/10.1080/00927872.2013.822878)", *Comm. Algebra* **42** (2014), 4854–4873 (=[arXiv:1305.0746](https://arxiv.org/abs/1305.0746)).

\[Freudenthal's paper is completely explicit and elementary; Fulton & Harris provide a bit more theoretical background; Faulkner is more general and works over an arbitrary commutative ring. —[Gro-Tsen](https://mathoverflow.net/users/17064/gro-tsen)\]


  [1]: http://www.ams.org/journals/bull/2016-53-04/S0273-0979-2016-01540-0/