Let R = _i ≥ 0R_i be a quadratic standard graded K-algebra. Backelin has shown that R is Koszul provided dim R₂ ≤ 2. One may wonder whether, under the same assumption, R is defined by a Gröbner basis of quadrics. In other words, one may ask whether an ideal I in a polynomial ring S generated by a space of quadrics of codimension ≤ 2 always has a Gröbner basis of quadrics. We will prove that this is indeed the case with, essentially, one exception given by the ideal I = (x², xy, y² − xz, yz)  K[x, y, z]. We show also that if R is a generic quadratic algebra with dim R₂ < dim R₁ then R is defined by a Gröbner basis of quadrics.