Let $L/K$ be a Galois extension of local fields of degree $n:=[L:K]<\infty$ with Galois group $G:=\operatorname{Gal}(L/K)$. In short, my question is as follows.
Is there an explicit computation of the fundamental class $u_{L/K}\in H^2\left(G,L^\times\right)$?
Here, the fundamental class is the generator of $H^2\left(G,L^\times\right)$ found by pulling back the generator $\frac1n\in\frac1n\mathbb Z/\mathbb Z$ along the invariant map $\operatorname{inv}_{L/K}:H^2\left(G,L^\times\right)\to\frac1n\mathbb Z/\mathbb Z$.
I record some thoughts below.
In the case where $L/K$ is unramified with Galois group $G:=\operatorname{Gal}(L/K)=\langle\operatorname{Frob}_{L/K}\rangle$, we have an explicit construction of the (local) invariant map $\operatorname{inv}_{L/K}:H^2\left(G,L^\times\right)\to\frac1n\mathbb Z/\mathbb Z$ by $$H^2\left(G,L^\times\right)\stackrel{\operatorname{ord}_L}\to H^2(G,\mathbb Z)\stackrel\delta\leftarrow H^1(G,\mathbb Q/\mathbb Z)\simeq\operatorname{Hom}_\mathbb Z(G,\mathbb Q/\mathbb Z)\stackrel{f\mapsto f(\operatorname{Frob}_{L/K})}\to\frac1n\mathbb Z/\mathbb Z,$$ and then we can pull the (more or less canonical) generator $\frac1n$ back to the (canonical) generator $u_{L/K}\in H^2(G,L^\times),$ which is called the fundamental class. All of this computation can be made explicit (see, for example the discussion preceding Proposition III.1.9 of Milne's notes); for completeness, we record that the computation gives $$u_{L/K}\left(1,\operatorname{Frob}_{L/K}^k,\operatorname{Frob}_{L/K}^{k+\ell}\right)=\begin{cases} \varpi & k+\ell\ge n, \\ 1 & k+\ell<n, \end{cases}$$ where $\varpi$ is a uniformizer of $L.$
In the general case (i.e., dropping the assumption that $L/K$ is unramified), the invariant map is not so explicit. Looking the proofs I've found around, the most explicit construction comes from writing the exact sequence $$0\to H^2\left(\operatorname{Gal}(L/K),L^\times\right)\to H^2\left(\operatorname{Gal}(K^{\text{unr}}/K),K^{\text{unr}\times}\right)\to H^2\left(\operatorname{Gal}(L^{\text{unr}}/L),L^{\text{unr}\times}\right)$$ (It takes some amount of work to define the maps and check that this is in fact a short exact sequence; it is essentially Lemma III.2.2 in Milne's notes.) In theory, we could say that $u_{L/K}$ will be the pull-back of $\operatorname{inv}_{K^{\text{unr}}/K}^{-1}\left(1/n\right)$ to $H^2(G,L^\times)$. However, I have been unable to make this computation explicit.
Actually, for a finite galois extension L/K of local fields with group G, the calculation of the fundamental class $u_{L/K} \in H^2(G, L^*)$ is tantamount to the determination of the so called reciprocity isomorphism of CFT, $\omega: K^*/N(L^*) \cong G^{ab}$, where N is the norm map and $G^{ab}$ the abelianized quotient of G, see e.g. Serre's book "Local Fields", chap. XIII, §4. More precisely, the cup-product by $u_{L/K}$ defines an isomorphism $\hat H^{-2}(G,Z) \cong \hat H^0(G,L^*)$, and $\omega$ is the inverse isomorphism. Notation: if $x \in L^*$ has image $\bar x$ in $K^*/N(L^*)$, write $(x, L/K) = \omega (\bar x)$ and call it the norm residue symbol.
When $L/K$ is unramified, if the Galois group of $L/K$ is identified with that of the residual extension $l/k$, it is a simple matter of book keeping to show that $(x, L/K)= (Frob_{k})^{v(x)}$ and check that this gives your formula for the fundamental class $u_{L/K}$ (Serre, prop. 13). In the general case, we must naturally pass through the totally unramified sub-extension $M$ contained in $L$ and work with the totally ramified $L/M$. The last step requires a deep new result due to Dwork (1958), which is explained in detail in Serre, thm. 2. Since the complete notations and results are complicated, allow me to cite only the corollary which will be used. So let $L/M$ be totally ramified, galois with group $G$; if $M_{nr}$ is the maximal unramified extension of $M$, we know that $L_{nr}:=L.M_{nr}$ is galois over $M_{nr}$ with group $G$, and the same is true for the extension of completed fields $\hat L_{nr}/ \hat M_{nr}$. The corollary states that, for $x \in M^{*}, y \in \hat L_{nr}^{*}, s \in G, \pi$ an uniformizer of $\hat L_{nr}$ such that $x=Ny$ and $F(y)/y = s(\pi)/(\pi)$ (here $F$ is the canonical generator of $G$ corresponding to 1 when $G$ is identified with $\hat Z$, and the existence of $x, y$ is due to the finiteness of the residual field). Then $(x, L/M) = s^{-1}$. This determines the norm residue symbol of $L/M$, hence also the fundamental class .
NB: In the "general case", it would be futile to look for more "explicit formulas", because of course more generality implies less precision, or requires more datas.