Two points: (1)The expression $q(x_1,\cdots,x_n)=\sum_{i,j} a_{ij}x_ix_j$ is ambiguous in that it does not define $a_{ij}$ uniquely, as Andrea said. One of the matrices $A=(a_{ij})$ which determines $q$ in this sense (and the only symmetric one which does this) is the matrix of the associated bilinear form in the same basis. We can indeed characterize singular quadratic forms as those whose matrix (in this sense) is singular, in any (equivalently, in some) basis of the vector space. (2) Why do we define a "singular" (or "degenerate", which I'm more used to) quadratic form in terms of its associated bilinear form, rather than simply going to the associated matrix? Well, mainly because in Linear Algebra we'd rather have intrinsic definitions whenever they are at our disposal. That is, if some concept can be characterized in a coordinate-free way (without having to express the object at hand in terms of any basis of the vector space), we'll prefer these definitions.