There isn't a way to define it that helps to formulate the law of quadratic reciprocity, which is what Legendre was trying to do when introducing his symbol.
As I'm sure you observed, every $a$ is a quadratic residue mod 2, but 2 is not a quadratic residue modulo every modulus, or modulo every prime modulus. So computing when 2 is a quadratic residue mod $p$ is interesting (for odd primes $p$), but computing when $p$ is a quadratic residue mod 2 is trivial.
Added: If we wanted to define the Legendre symbol for $p=2$, just for the sake of completeness, the natural definition would be $(a/p) = 1$ for $a$ odd and $(a/p) = 0$ for $a$ even. It's important that we keep in mind that quadratic reciprocity only works with two odd primes, so omitting $p=2$ from the definition of the Legendre symbol is justifiable for avoiding any confusion on that point. What this does "make work" is the multiplicative character of the symbol, i.e. $(ab/p) = (a/p)(b/p)$ now also when $p=2$.