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Gas giants are the most readily detectable exoplanets but fundamental questions about their system architectures, formation, migration, and atmospheres have been unanswerable with the current generation of ground- and space-based facilities. The dominant techniques to detect and characterize giant planets $-$ radial velocities, transits, direct imaging, microlensing, and astrometry $-$ are each isolated to a limited range of planet masses, separations, ages, and temperatures. These windows into the arrangement and physical properties of giant planets have spawned new questions about the timescale and location of their assembly; the distributions of planet mass and orbital separation at young and old ages; the composition and structure of their atmospheres; and their orbital and rotational angular momentum architectures. The ELTs will address these questions by building bridges between these islands of mass, orbital distance, and age. The angular resolution, collecting area, all-sky coverage, and novel instrumentation suite of these facilities are needed to provide a complete map of the orbits and atmospheric evolution of gas giant planets (0.3$-$10 $M_\mathrm{Jup}$) across space (0.1$-$100 AU) and time (1 Myr to 10 Gyr). This white paper highlights the scientific potential of the GMT and TMT to address these outstanding questions, with a particular focus on the role of direct imaging and spectroscopy of large samples of giant planets that will soon be made available with $Gaia$., White paper for the Astro2020 decadal survey