What is the matrix of the monster Lie algebra? In Richard Borcherds' proof of monstrous moonshine, he constructs a "monster Lie algebra", which is a $\mathbb Z^2$-graded, infinite-dimensional Lie algebra with a contravariant bilinear form acted on by the monster group. The monster Lie algebra is a generalized Kac-Moody algebra, so is associated to a infinite symmetric "matrix" $A = (a_{ij})_{ij}$ satisfying certain conditions.

Is a recurrence relation (or an explicit formula) for the $a_{ij}$ known?
I guess your difficulty might be that in the papers, the root spaces are illustrated without telling you the GKM matrix coefficients. Actually these coefficients can be obtained directly from the basic properties of the monster Lie algebra, and it depends on how much you know about the construction. Indeed the whole matrix can be explicitly described using the coeffficients of the (normalized) $j$-function, which is well known (at least to experts). If you feel it hard to work it out yourself, the direct pictorial answer can be seen in Roger Carter's book, or p.22--23 of this article: 

http://jurisiche.people.cofc.edu/jur696.pdf 

for free. Besides the original papers by Borcherds, the articles by Elizabeth Jurisich and the book by Urmie Ray will tell more details.
I guess your difficulty might be that in the papers, the root spaces are illustrated without telling you the GKM matrix coefficients. Actually these coefficients can be obtained directly from the basic properties of the monster Lie algebra, and it depends on how much you know about the construction. Indeed the whole matrix can be explicitly described using the coeffficients of the (normalized) $j$-function, which is well known (at least to experts). If you feel it hard to work it out yourself, the direct pictorial answer can be seen in Roger Carter's book, or p.22--23 of this article: 

http://jurisiche.people.cofc.edu/jur696.pdf 

for free. Besides the original papers by Borcherds, the articles by Elizabeth Jurisich and the book by Urmie Ray will tell more details.