Is there a succinct representation for all problems?
Any language $L$ consisting of strings of length $2^k$ for nonnegative integers k has a succinct version: given a binary circuit $C$ that takes $k$ inputs and outputs a single bit, is the truth table (seen as a binary string of length $2^k$) a string of the language $L$. You can restrict any language to strings whose length is a power of two, or you can try other encoding tricks.