Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as $\text{Mat}_n(k)$ but considered as a Lie algebra. We can form the universal enveloping algebra $U(gl_n(k))$ of $gl_n(k)$, which satisfies the universal property, $\forall$ unital associative $k$-algebra $A$, we have $$ Hom_{k-alg}(U(gl_n(k)),A)\cong Hom_{Lie}(gl_n(k),F(A)), $$ where $F$ is the forgetful functor from $k$-alg to Lie.

Now we know that the associative algebra $\text{Mat}_n(k)$ is not the universal enveloping algebra $U(gl_n(k))$. However, by the universal property, we have $$ Hom_{k-alg}(U(gl_n(k)),\text{Mat}_n(k))\cong Hom_{Lie}(gl_n(k),gl_n(k)), $$ which makes $\text{Mat}_n(k)$ "special" in some sense.

$\textbf{My question}$ is: Could we make the above observation more precise? Does $\text{Mat}_n(k)$ have some universal properties similar to $U(gl_n(k))$?