We study convergence of multilinear forms $\sum f(n_1,\ldots, n_k) X_{n_1} \cdots X_{n_k}$ in symmetric independent random variables. We show that the multilinear form converges if and only if its "tetrahedronal" part and "diagonal" parts of different orders converge simultaneously. For "tetrahedronal" forms a.s. and $L_0$ convergence are equivalent. Moreover, they are equivalent to $L_p$ convergence provided $(X_k)$ satisfies a Marcinkiewicz-Paley-Zygmund condition for $p \geq 2$.