= Solution
Let $\pi:T^*L\to L$ be the <cotangent bundle> projection. The <canonical one-form on a cotangent bundle> is intrinsically defined by
$$
\lambda_{(q,p)}(V)=p(d\pi(V)).
$$
No metric or coordinate choice is needed. In local coordinates $\lambda=\sum_i p_i\,dq^i$. Choose
$$
\boxed{\omega_{\mathrm{can}}=-d\lambda=\sum_i dq^i\wedge dp_i.}
$$
It is closed because $d^2=0$, and its coordinate matrix is $\begin{pmatrix}0&I\\-I&0\end{pmatrix}$, which is invertible. The intrinsic definition of $\lambda$ makes the forms agree under all cotangent coordinate changes. This is the canonical <symplectic form>; choosing $d\lambda$ instead is the opposite common sign convention.
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