I tried to just comment, but it seems that I do not have enough "reputation" for that. So I am commenting here.

The answer would be "no" if your field were infinite. Since in that case, each homogeneous component of a polynomial identity would also be an identity (see [1], prop. 4.2.3). And that would imply that you would have a multilinear identity of the same degree of such monomial (use the multilinearization process (see [1])).
Finally, it is easy to show that there does not exist polynomial identities of degree less than 2n for M_n(F) (staircase argument see [1] Exercise 7.1.2 - this is true for any field).

Ok, but it was not asked about infinite fields.

For matrices over finite fields, there are descriptions for bases of identities for matrices of order up to 4. (see [2,3,4])
I checked the papers for 2 and 3 (unfortunately the case $n=4$ is in russian), and some of the generators of the ideal of identities of such algebras are polynomials which contains monomials of degree exactly n (2 or 3, in this case), but not less than n. So, it does not answer your question.

At first sight I thought the answer to your question would be "no" even if you were asking for the degree of your monomial to have degree less than 2n. For 2n, the papers [2] and [3] show I was wrong.

References:

[1] V. Drensky, Free Algebras and PI-algebras: Graduate Course in Algebra, Springer, Singapore, 1999.

[2] Yu. N. Mal'tsev, E. N. Kuz'min, A basis for the identities of the algebra of second-order matrices over a finite field, Algebra Log. 17 (1978) 18–21

[3] G. Genov, Basis for identities of a third order matrix algebra over a finite field, Algebra Log. 20 (1981) 241–257.

[4] G. Genov, P. Siderov, A basis for identities of the algebra of fourth-order matrices over a finite field. I, II, Serdica 8 (1982) 313–323, 351–366 (in Russian).