How to describe the compact real forms of the exceptional Lie groups as matrix groups?

Cartan describes all of the compact real forms of the simple Lie groups over $\mathbb{C}$ in his first paper that classifies the real forms. In fact, he describes them exactly in the terms that you ask for: A representation of the complex Lie group together with an auxilliary structure, either a real structure on the complex representation space or a Hermitian quadratic form.

For $\mathrm{G}_2$ (resp, $\mathrm{F}_4$, $\mathrm{E}_8$), the compact forms are represented as special orthogonal real matrices of rank 7 (resp., 26, 248). For $\mathrm{E}_6$ (resp. $\mathrm{E}_7$), the compact real forms are represented as special unitary matrices of rank $27$ (resp. $56$).

Explicitly, here are the defining structures in the lowest dimensional representations of the compact real forms of the exceptional groups:

• $\mathrm{G}_2$ is the stabilizer of a $3$-form on a real vector space of dimension $7$.

• $\mathrm{F}_4$ is the stabilizer of a quadratic form and a cubic form on a real vector space of dimension $26$. (I believe that the cubic form alone is enough to define $\mathrm{F}_4$.)

• $\mathrm{E}_6$ is the stabilizer of a cubic form and a positive definite Hermitian form on a complex vector space of dimension $27$. (The cubic form alone only defines the complex $\mathrm{E}_6$.)

• $\mathrm{E}_7$ is the stabilizer of a symplectic form, a quartic form, and a positive definite Hermitian form on a complex vector space of dimension $56$. (The quartic form and the Hermitian form by themselves are almost enough to define $\mathrm{E}_7$; they define a group with two connected components, the identity component of which is the compact $\mathrm{E}_7$.)

• $\mathrm{E}_8$ is the stabilizer of a $3$-form on a real vector space of dimension $248$.

There is an abstract way of integrating Lie algebras but I guess you are asking for a more hands on approach. I suggest browsing Exceptional Lie groups by Ichiro Yokota. Usually, it's the compact (or perhaps complex) Lie groups which are treated in the literature, so maybe you should be more specific in what do you think is lacking there.

As far as $F_4$ goes the description can be made rather succinct using so called Jordan algebra:

Take the real vector space of octonionic-Hermitian three by three matrices and endow it with commutative product defined by $A \circ B = \frac{1}{2}(AB+BA).$ The automorphism group of this product is the compact Lie group of type $F_4.$ One can actually define some kind of octonionic determinant for this Jordan algebra and then it can be proved that $F_4$ is the group stabilizing this determinant and trace. If you take just the group stabilizing the determinant, you will obtain noncompact real form of $E_6.$