Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers? Let $G$ be a finite abelian $p$-group (where $p$ is a prime). Suppose there exists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle g,\;\rangle$, is an isomorphism from $G$ to $\mathrm{Hom} (G, \mathbb{Q}/\mathbb{Z})$.
Then, is it true that $G$ can be written as $\mathbf{Z}_p^n/\,\mathrm{im} (A)$ for some symmetric matrix (i.e. a linear map) $A\colon\mathbf{Z}_p^n\to\mathbf{Z}_p^n$ for some positive integer $n$? (Here $\mathbf{Z}_p$ denotes the $p$-adic integers)
EDIT2: As user74230 has pointed out, the isomorphism $G\to \mathbf{Z}_p^n/\,\mathrm{im} (A)$ has to respect the evident bilinear forms on both sides. (Otherwise, the answer is trivial and $A$ does not depend on $\delta$ as Amritanshu Prasad has pointed out).
EDIT1: I had this question while reading the following paper: http://arxiv.org/pdf/1402.5129v1. In Theorem 2 of page 4 it says that, given a $p$-group with duality pairing $(G,\delta )$ the probability (w.r.t. Haar measure) that $\mathrm{coker}(A)\cong (G,\delta)$ converges to Cohen-Lenstra type probability measure when $n\to\infty$. The existance of the duality pairing seems relevant for this theorem, but Amritanshu Prasad's comment suggests that $A$ can be chosen independent of the pairing which seems puzzling for me.
 A: For odd $p$ the answer is affirmative. Suppose $p$ is odd, so $\delta$ is valued in $\mathbf{Q}_p/\mathbf{Z}_p$ and we may define the quadratic form $q:G \rightarrow \mathbf{Q}_p/\mathbf{Z}_p$ by $q(g) = (1/2)\delta(g,g)$ so that $(G,q)$ is a "non-degenerate" finite quadratic space with associated symmetric bilinear form $\delta$. 
Your question for such $p$ amounts to asking if every non-degenerate finite quadratic space of $p$-power size is the "discriminant form" of a quadratic lattice over $\mathbf{Z}_p$.  This is part of the assertion of Theorem 1.9.1 of Nikulin's paper "Integral symmetric bilinear forms and some of their applications" in Math. USSR Izv. Vol. 14 No. 103 (1980), which also gives uniqueness aspects (under a suitable minimality requirement) and also provides analogues when $p=2$. The result there for $p=2$ probably also solves your question for $p=2$ but it would require some care to unravel the passage between bilinear forms and quadratic forms in the 2-adic setting.
Nikulin's proof amounts to the systematic study of primary parts under orthogonality and inductive knowledge built from the classification of non-degenerate quadratic spaces over finite fields.
