Given an almost complex structure $J:TM\to TM$, the Nijenhuis tensor $N_J:\wedge^2TM\to TM$ is given by $N_{J}(X,Y)=[X,Y]+J([JX,Y]+[X,JY])-[JX,JY]$.
My question is, is there a necessary and sufficient condition (some algebraic identities) for a (1,2) tensor $N$, such that locally it must equals to $N_J$ for some $J$?
I was hoping to find some conditions on certain $(1,3)$ tensors induced by $N$.
Yes, there are pointwise algebraic conditions on a section $N$ of $T\otimes\Lambda^2(T^*)$ in order for $N$ to equal $N_J$ for some almost complex structure $J$, but there are differential conditions that must be satisfied as well.
If $M$ has (real) dimension $2n>2$ and has almost complex structures (which not every $2n$-manifold does) then each $N_J$ takes values in an algebraic conical subset $C\subset T\otimes\Lambda^2(T^*)$ whose fiber $C_x\subset T_x\otimes\Lambda^2(T^*_x)$ over each point $x\in M$ is a (singular) cone of dimension $n^2(n{+}1)$ inside the vector space $T_x\otimes\Lambda^2(T^*_x)$ (which has dimension $n^2(4n{-}2)$). (Of course, when $n=1$, $C$ consists of the zero section of $T\otimes\Lambda^2(T^*)$.)
In fact, it's not difficult to show that $C$ actually lies inside the bundle $K\subset T\otimes\Lambda^2(T^*)$ that is the kernel of the natural contraction $T\otimes\Lambda^2(T^*)\to T^*$, but, when $n>1$, $C$ is a proper subset of $K$, so requiring $N_J$ to lie in $K$ is not enough when $n>1$.
In the first nontrivial case, $n=2$, one has an easy characterization: If $N$ is a nonvanishing section of $T\otimes\Lambda^2(T^*)$ that equals $N_J$ for some $J$, then one finds that $N$ must be a section of $T\otimes E$, where $E\subset\Lambda^2(T^*)$ is a subbundle of rank $2$ that contains no nonzero decomposable elements. Thus, locally there will be a basis of $1$-forms $\eta^1,\ldots,\eta^4$ so that $$ N = X_1\otimes(\eta^1\wedge\eta^3-\eta^2\wedge\eta^4) + X_2\otimes(\eta^1\wedge\eta^4-\eta^3\wedge\eta^2) $$ where $X_1$ and $X_2$ are nonzero vector fields. Moreover, the trace condition (i.e., $N$ must be a section of $K$) implies $\eta^1(X_1)+\eta^2(X_2)=0$ and three further linear equations, showing that $X_2$ is determined by $X_1$ and vice versa. It then follows that either $J$ or $-J$ must be the almost complex structure for which $\zeta^1=\eta^1+\mathrm{i}\,\eta^2$ and $\zeta^2=\eta^3+\mathrm{i}\,\eta^4$ are the $(1,0)$-forms. Thus, computing $N_J=N_{-J}$ for either of these particular $J$s, one sees that, if this $N_J$ is not equal to $N$, then $N$ is not a Nijnhuis tensor. This last test is applied by taking one derivative of the $J$ that was determined algebraically, so it constitutes $4$ first-order PDE on $N$.
There are similar but more complicated necessary and sufficient conditions for higher values of $n$.
Addendum: In order to verify the above claims, it might be useful to have a slightly different formulation of the Nijnhuis tensor that is more computation friendly and that will make some of the claims above relatively transparent, so here it is:
Fix an almost complex structure $J$ on $M^{2n}$ and let $U\subset M$ be an open set on which there exists a basis $\zeta^1,\ldots,\zeta^n$ for the $(1,0)$-forms of $J$ on $U$. Let $Z_1,\ldots Z_n$ be the sections of $T\otimes\mathbb{C}$ that satisfy $\zeta^j(Z_k) = \delta^k_j$ while $\overline{\zeta^j}(Z_k) = 0$. (N.B.: If $X$ is a (real) vector field on $M$ and $\omega$ is a $\mathbb{C}$-valued $1$-form on $M$, then, by definition, $\omega(\mathrm{i}\,X) = \mathrm{i}\,\omega(X)$.) Then, up to a universal constant multiple, $$ N_J = Z_k\otimes(\mathrm{d}\zeta^k)^{(0,2)} + \overline{Z_k}\otimes \overline{(\mathrm{d}\zeta^k)^{(0,2)}}, $$ where, for a $\mathbb{C}$-valued $2$-form $\tau$, the notation $\tau^{(0,2)}$ denotes the $(0,2)$-component of $\tau$ with respect to $J$. Thus, since $(\mathrm{d}\zeta^k)^{(0,2)} = \tfrac12\, A^k_{\bar p\bar q}\,\overline{\zeta^p}\wedge\overline{\zeta^q}$ for some functions $A^k_{\bar p\bar q} = -A^k_{\bar q\bar p}$, we have $$ N_J = \mathrm{Re}\left(A^k_{\bar p\bar q}\,Z_k\otimes \overline{\zeta^p}\wedge\overline{\zeta^q} \right). $$ There are no further pointwise constraints on the complex functions $A^k_{\bar p\bar q}$, so, at $x\in M$, once $J_x$ is fixed, there is a vector space of real dimension $n^2(n{-}1)$ of possible values of $N_J$ at $x$. Since there is a $2n^2$-parameter family of possible values of $J_x$, it follows without much difficulty that the conical subset $C_x$, which is a union of a $2n^2$-parameter family of subspaces, each of dimension $n^2(n{-}1)$, has dimension $n^2(n{+}1)$. It also follows from the above formula that $N_J$ is a section of $K$, since $\overline{\zeta^j}(Z_k) = 0$.
