Let $V$ be a vector space with a basis $v\_1, v\_2, \ldots, v\_n$. Let $T(V)$ be the tensor algebra of $V$. Let $S(Lie(V))$ be the symmetric algebra of the free Lie algebra of $V$. I think that $T(V)$ is isomorphic to $S(Lie(V))$ as a graded vector space. For example, let $n=2$. We have the degree $2$ component of $T(V)$ is
\begin{align}
 T(V)\_2 = Span\{v\_1 \otimes v\_1, v\_1 \otimes v\_2, v\_2 \otimes v\_1, v\_2 \otimes v\_2\}.
\end{align}
The degree 2 component of $S(Lie(V))$ is
\begin{align}
 S(Lie(V))\_2 = Span\{v\_1^2, v\_2^2, v\_1 v\_2, [v\_1, v\_2]\}.
\end{align}
A bijection from $T(V)\_2$ to $S(Lie(V))\_2$ is
\begin{align}
& v\_1 \otimes v\_1 \mapsto v\_1^2, \\
& v\_1 \otimes v\_2 \mapsto v\_1 v\_2, \\
& v\_2 \otimes v\_1 \mapsto [v\_1, v\_2], \\
& v\_2 \otimes v\_2 \mapsto v\_2^2.
\end{align}


The degree $3$ component of $T(V)$ is
\begin{align}
& T(V)\_3 = Span\{v\_1 \otimes v\_1 \otimes v\_1, v\_1 \otimes v\_1 \otimes v\_2, v\_1 \otimes v\_2 \otimes v\_1, v\_2 \otimes v\_1 \otimes v\_1, \\
& \quad \quad \quad \quad \quad v\_1 \otimes v\_2 \otimes v\_2, v\_2 \otimes v\_1 \otimes v\_2, v\_2 \otimes v\_2 \otimes v\_1, v\_2 \otimes v\_2 \otimes v\_2 \}.
\end{align}
The degree 2 component of $S(Lie(V))$ is
\begin{align}
 S(Lie(V))\_2 = Span\{v\_1^3, v\_2^3, v\_1^2 v\_2, v\_1 v\_2^2, v\_1 [v\_1, v\_2], v\_2[v\_1,v\_2], [[v\_1,v\_2],v\_2], [[v\_1,v\_2],v\_1]\}.
\end{align}
A bijection from $T(V)\_3$ to $S(Lie(V))\_3$ is
\begin{align}
& v\_1 \otimes v\_1 \otimes v\_1 \mapsto v\_1^3, \\
& v\_2 \otimes v\_2 \otimes v\_2 \mapsto v\_2^3, \\
& v\_1 \otimes v\_2 \otimes v\_2 \mapsto v\_1 v\_2^2, \\
& v\_1 \otimes v\_1 \otimes v\_2 \mapsto v\_1^2 v\_2, \\
& v\_1 \otimes v\_2 \otimes v\_1 \mapsto v\_1 [v\_1, v\_2], \\
& v\_2 \otimes v\_1 \otimes v\_2 \mapsto [v\_1, v\_2]v\_2, \\
& v\_2 \otimes v\_2 \otimes v\_1 \mapsto [[v\_1,v\_2],v\_2], \\
& v\_2 \otimes v\_1 \otimes v\_1 \mapsto [[v\_1, v\_2], v\_1].
\end{align}
How to write down the general formula $T(V)\_n \to S(Lie(V))\_n$?



How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly?
The natural map is rather from $TV \to U(FreeLie(V))$: consider the forgetful functors $Assoc \to Lie \to Vec$ and compose their left adjoints to get the left adjoint $T$ of the composite. Then, as Alex Suciu comments, your question is a special case of the comparison of $U\mathfrak g$ and $S\mathfrak g$, accomplished by PBW.


