The answer to your question is "no", as explained by Anton Klyachko in his answer. Let me refer you to a remarkable statement of Razmyslov and Procesi that describes all identities. They proved (independently) that in fact all identities of $Mat_n(\mathbb{Q})$ follow, in a sense, from the Cayley--Hamilton theorem. To be more precise:

Let us first define the notion of an *algebra with trace* as a vector space $A$ over a field $F$ that has a bilinear product, and a linear functional $t\colon A\to F$ satisfying $t(AB)=t(BA)$. Of course we can consider the free algebra with trace generated by a set $X$; it is the free associative algebra generated by $X$ over the ring of polynomials in $t(m)$, where $m$ is a cyclic word in $X$ (a cyclic group orbit on words), and $t$ is extended to this algebra by an obvious rule $t(t(m_1)m_2)=t(m_1)t(m_2)$. This gives us a language to discuss we can talk about identities of algebras with trace. (Alternatively, one can use the language of operads to discuss that; I prefer that latter language but choose to write a more "classical" definition here).

Next, let us define the $n$-th Cayley-Hamilton identity of an algebra with trace as the identity
$$
CH_n=\sum_{\sigma\in S_{n+1}}X_{i_1^{(1)}}\cdots X_{i_{k_1}^{(1)}}t(X_{i_1^{(2)}}\cdots X_{i_{k_2}^{(2)}})\cdots t(X_{i_1^{(l)}}\cdots X_{i_{k_l}^{(l)}})=0
$$
where $\sigma$ has the disjoint cycle decomposition $(0,i_1^{(1)},\ldots,i_{k_1}^{(1)})(i_1^{(2)},\ldots,i_{k_2}^{(2)})\cdots(i_1^{(l)},\ldots,i_{k_l}^{(l)})$. (This is a full multilinearisation of the Cayley--Hamilton theorem $\chi_A(A)=0$).

Theorem (Procesi [1], Razmyslov[2]). Every identity of the matrix algebra viewed as an algebra with trace is a consequence of the identity $CH_n=0$.

[1] C.Procesi, The invariant theory of n × n matrices, Advances in Mathematics
Volume 19, Issue 3, March 1976, Pages 306–381,
http://www.sciencedirect.com/science/article/pii/000187087690027X

[2] Yu. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero,
Izv. Akad. Nauk SSSR Ser. Mat., 1974, Volume 38, Issue 4, Pages 723–756, http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1989&option_lang=eng

Of course, every usual identity is a particular case of a trace identity, so in principle this theorem also classifies all usual identities.