In a mixing bowl, combine the soy sauce, brown sugar, sesame oil, minced garlic, grated ginger, rice vinegar, and red pepper flakes. Whisk together until the sugar has dissolved and the ingredients are well combined.
Place the salmon fillets in a baking dish and pour the marinade over them. Ensure each fillet is well coated. Cover and let the salmon marinate in the refrigerator for at least 30 minutes. If you have more time, let it marinate for up to two hours for enhanced flavor.
Preheat your oven to 400°F (200°C) or prepare your grill for medium-high heat.
Remove the salmon from the marinade and place it skin-side down on a baking sheet lined with parchment paper, or directly on the grill. Cook for about 12-15 minutes, or until the salmon is flaky and cooked through. If you're using an oven, you can broil it for the last 2-3 minutes for a slightly charred effect.
While the salmon is cooking, you can make a simple dipping sauce using the leftover marinade. In a small saucepan, bring the marinade to a boil and let it simmer for 2-3 minutes to thicken slightly. Remove from heat and allow to cool.
Once the salmon is cooked, remove it from the oven or grill. Drizzle the dipping sauce over the salmon and garnish with sliced green onions and sesame seeds. Serve with steamed rice and vegetables for a complete meal.