There are several well known presentations. One has to first choose your favorite presentation of the symmetric group. Then one usually adds in a rank n-1 idempotent but some people add all of them or other elements. Then one figures out the extra relations. See page 161 of Classical Finite Semigroups by Ganyushkin and Mazorchuk. You should know the monoid of all self-maps of a set is usually called the full transformation monoid or $T_n$ when googling.
Pages 42-43 of http://research-repository.st-andrews.ac.uk/bitstream/10023/2821/3/NikolaRuskucPhDThesis.pdf give two presentations.